(a) Find the particular solution of the differential equation $\frac{dy}{dx} - 2xy = 3x^2 e^{x^2} ; y(0) = 5$. OR (b) Solve the following differential equation : $x^2 dy + y(x + y) dx = 0$
CBSE CLASS 12 • MATHEMATICS • CHAPTER 09
Differential Equations
195 distinct questions across 224 board paper set appearances (2022–2026).
The integrating factor of the differential equation $(x + 2y^2) \frac{dy}{dx} = y$ $(y > 0)$ is :
(A) $\frac{1}{x}$
(B) $x$
(C) $y$
(D) $\frac{1}{y}$
The integrating factor of the differential equation $(1 - y^2) \frac{dx}{dy} + yx = ay$, $(-1 < y < 1)$ is
(A) $\frac{1}{y^2 - 1}$
(B) $\frac{1}{\sqrt{y^2 - 1}}$
(C) $\frac{1}{1 - y^2}$
(D) $\frac{1}{\sqrt{1 - y^2}}$
Which of the following is not a Linear Differential Equation ?
(A) $(1 + x^2) dy + 2xy dx = \cot x dx$
(B) $y + \frac{d}{dx}(xy) = x (\sin x + \log x)$
(C) $x (1 + y^2) dx - y (1 + x^2) dy = 0$
(D) $y dx - (x + 3y^2) dy = 0$
(a) Solve the differential equation $y e^y dx = (y^3 + 2x e^y) dy$, when $y(0) = 1$.
OR
(b) Find the general solution of the differential equation $(x^3 - 3xy^2) dx = (y^3 - 3x^2y) dy$.
If 'm' and 'n' are the degree and order respectively of the differential equation $1 + \left(\frac{\text{dy}}{\text{dx}}\right)^3 = \frac{\text{d}^2\text{y}}{\text{dx}^2}$, then the value of $(m + n)$ is :
(A) 4
(B) 3
(C) 2
(D) 5
The integrating factor of the differential equation $(1 - x^2) \frac{dy}{dx} + xy = ax$, $-1 < x < 1$, is :
(A) $\frac{1}{x^2 - 1}$
(B) $\frac{1}{\sqrt{x^2 - 1}}$
(C) $\frac{1}{1 - x^2}$
(D) $\frac{1}{\sqrt{1 - x^2}}$
The sum of the order and degree of the differential equation $\left[1+\left(\frac{\text{d}y}{\text{d}x}\right)^2\right]^3 = \frac{\text{d}^2y}{\text{d}x^2}$ is :
(A) $2$
(B) $\frac{5}{2}$
(C) $3$
(D) $4$
(a) Solve the differential equation : $x^2y \, dx - (x^3 + y^3) \, dy = 0$.
OR
(b) Solve the differential equation $(1 + x^2) \frac{dy}{dx} + 2xy - 4x^2 = 0$ subject to initial condition $y(0) = 0$.
Camphor is a waxy, colourless solid with strong aroma that evaporates through the process of sublimation, if left in the open at room temperature. A cylindrical camphor tablet whose height is equal to its radius (r) evaporates when exposed to air such that the rate of reduction of its volume is prop...
Write the order and degree of the given differential equation.
Substituting $V = \pi r^3$ and $S = 2\pi r^2$, we get the differential equation $\frac{dr}{dt} = \frac{2}{3}k$. Solve it, given that $r(0) = 5\text{ mm}$.
(a) Find the particular solution of the differential equation given by $2xy + y^2 - 2x^2 \frac{dy}{dx} = 0$; $y = 2$, when $x = 1$. OR (b) Find the general solution of the differential equation : $y dx = (x + 2y^2) dy$
A bacteria sample of certain number of bacteria is observed to grow exponentially in a given amount of time. Using exponential growth model, the rate of growth of this sample of bacteria is calculated. The differential equation representing the growth of bacteria is given as : $\frac{\text{dP}}{\tex...
(i) Obtain the general solution of the given differential equation and express it as an exponential function of 't'.
x \log x \frac{dy}{dx} + y = 2 \log x is an example of a :
(A) variable separable differential equation.
(B) homogeneous differential equation.
(C) first order linear differential equation.
(D) differential equation whose degree is not defined.
The general solution of the differential equation $x \, dy - (1 + x^2) \, dx = dx$ is :
(a) $y = 2x + \frac{x^3}{3} + C$
(b) $y = 2 \log x + \frac{x^3}{3} + C$
(c) $y = \frac{x^2}{2} + C$
(d) $y = 2 \log x + \frac{x^2}{2} + C$
(a) Find the general solution of the differential equation :
$(xy - x^2) dy = y^2 dx.$
OR
(b) Find the general solution of the differential equation :
$(x^2 + 1) \frac{dy}{dx} + 2xy = \sqrt{x^2 + 4}$
(a) Find the general solution of the differential equation : $\frac{d}{dx}(xy^{2}) = 2y(1 + x^{2})$ OR (b) Solve the following differential equation : $xe^{\frac{y}{x}} - y + x \frac{dy}{dx} = 0$
The integrating factor for solving the differential equation $x \frac{dy}{dx} - y = 2x^2$ is :
(a) $e^{-y}$
(b) $e^{-x}$
(c) $x$
(d) $\frac{1}{x}$
(a) Find the particular solution of the differential equation $\frac{dy}{dx} = \frac{x + y}{x}$, $y(1) = 0$.
OR
(b) Find the general solution of the differential equation $e^x \tan y \, dx + (1 - e^x) \sec^2 y \, dy = 0$.
Find the particular solution of the differential equation $x \frac{dy}{dx} - y = x^2 \cdot e^x$, given $y(1) = 0$. OR Find the general solution of the differential equation $x \frac{dy}{dx} = y(\log y - \log x + 1)$.
Find the general solution of the differential equation $\sec^2 x \cdot \tan y \, dx + \sec^2 y \cdot \tan x \, dy = 0$.
Find the particular solution of the differential equation $x \frac{dy}{dx} + y + \frac{1}{1 + x^2} = 0$, given that $y(1) = 0$.
Find the general solution of the differential equation $x(y^3 + x^3) dy = (2y^4 + 5x^3y) dx$.
The order and degree of the differential equation $\\frac{\mathrm{d}}{\mathrm{d}x}(e^y) = 0$ respectively are
(A) $0, 1$
(B) $1, 1$
(C) $2, 1$
(D) $1, \text{not defined}$
(a) Find the general solution of the following differential equation : \
$x^2 \\frac{\text{d}y}{\text{d}x} = x^2 + xy + y^2$ \
OR \
(b) Find the particular solution of the differential equation \
$xy \\frac{\text{d}y}{\text{d}x} = (x + 2)(y + 2)$, given that $y(1) = -1$.
The order and degree of the differential equation : $\frac{d}{dx} (\sin y) = y^2$ respectively are
(A) $1, 1$
(B) $2, 1$
(C) $2, 2$
(D) $1, 2$
(a) Find the general solution of the differential equation :
$y^2 \text{d}x + (x^2 - xy + y^2)\text{d}y = 0$
OR
(b) Find the particular solution of the differential equation $\frac{\text{d}y}{\text{d}x} = y \tan x$, given that $y = 2$ if $x = 0$.
The order and degree of the differential equation :
$\frac{d}{dx}(y')^{3} + (y')^{3} = 1$ respectively are where $y' = \frac{dy}{dx}$
(A) $1, 3$
(B) $2, 1$
(C) $3, 1$
(D) $3, 2$
(a) Find the general solution of the differential equation $(y^2 - x^2)\text{d}x = 2xy \text{d}y$\nOR\n(b) Find the particular solution of the differential equation $(1 + e^{2x})\text{d}y + (1 + y^2)e^x \text{d}x = 0$, given that $y(1) = 0$.
The general solution of the differential equation $\frac{dy}{dx} = \frac{\sqrt{y}}{\sqrt{x}}$ is
(A) $\log \sqrt{y} = \log \sqrt{x} + C$
(B) $\sqrt{y} + \sqrt{x} = C$
(C) $\sqrt{y} - \sqrt{x} = C$
(D) $\log \sqrt{y} + \log \sqrt{x} = C$
The integrating factor of the differential equation $2x \frac{\mathrm{d}y}{\mathrm{d}x} - y = 3$ is
(A) $\sqrt{x}$
(B) $\frac{1}{\sqrt{x}}$
(C) $e^x$
(D) $e^{-x}$
(a) Solve the following differential equation :
$x \frac{dy}{dx} = y - x \sin^2 \left(\frac{y}{x}\right)$, given that $y(1) = \frac{\pi}{6}$
OR
(b) Find the general solution of the differential equation : $y \log y \frac{dx}{dy} + x = \frac{2}{y}$.
The general solution of the differential equation $x \, dy - y \, dx = 0$ is
(A) $x^2 - y^2 = k$
(B) $xy = k$
(C) $x = ky$
(D) $\log y + \log x = k$
The integrating factor of the differential equation $2x \frac{dy}{dx} - y = 3$ is
(A) $\sqrt{x}$
(B) $\frac{1}{\sqrt{x}}$
(C) $e^x$
(D) $e^{-x}$
(a) Solve the following differential equation :
$x \frac{\text{d}y}{\text{d}x} = y - x \sin^2\left(\frac{y}{x}\right)$, given that $y(1) = \frac{\pi}{6}$
(b) Find the general solution of the differential equation : $y \log y \frac{\text{d}x}{\text{d}y} + x = \frac{2}{y}$.
The general solution of the differential equation : $x^2\text{d}y + y^2\text{d}x = 0$ is
(A) $x^3 + y^3 = \text{k}$
(B) $\frac{1}{y} - \frac{1}{x} = \text{k}$
(C) $\frac{1}{y} + \frac{1}{x} = \text{k}$
(D) $\log y^2 + \log x^2 = \text{k}$
The integrating factor of the differential equation $2x\frac{\text{d}y}{\text{d}x} - y = 3$ is
(A) $\sqrt{x}$
(B) $\frac{1}{\sqrt{x}}$
(C) $e^x$
(D) $e^{-x}$
(a) Solve the following differential equation :
$x \frac{\text{d}y}{\text{d}x} = y - x \sin^2 \left(\frac{y}{x}\right)$, given that $y(1) = \frac{\pi}{6}$
OR
(b) Find the general solution of the differential equation : $y \log y \frac{\text{d}x}{\text{d}y} + x = \frac{2}{y}$.
The general solution of the differential equation $\frac{\text{d}y}{\text{d}x} = e^{y - x}$ is :
(A) $e^x - e^y = C$
(B) $e^{-y} + e^x = C$
(C) $e^{-y} - e^{-x} = C$
(D) $e^{-x} + e^{-y} = C$
$\frac{dy}{dx} = \text{F}(x, y)$ will be a homogeneous differential equation for which of the following functions ?
(i) $\text{F}(x, y) = 3x + 2y$
(ii) $\text{F}(x, y) = \sin\left(\frac{y}{x}\right) + \log y - \log x$
(iii) $\text{F}(x, y) = e^{y/x} + 1$
(iv) $\text{F}(x, y) = \sqrt{x^2 + y^2} - y$
...
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is...
(a) Find the general solution of the differential equation $2x^2 \frac{dy}{dx} = y^2 + 2xy$.
OR
(b) Find a particular solution of the differential equation $(x+1) \frac{dy}{dx} = 2 e^{-y} - 1$, given that $y = 0$ when $x = 0$.
$\frac{\text{dy}}{\text{dx}} = \text{F}(x, y)$ will be a homogeneous differential equation for which of the following functions ?
(i) $\text{F}(x, y) = 3x + 2y$
(ii) $\text{F}(x, y) = \sin \frac{y}{x} + \log y - \log x$
(iii) $\text{F}(x, y) = e^{y/x} + 1$
(iv) $\text{F}(x, y) = \sqrt{x^2 + y^2} - y...
Assertion (A) : One of the particular solutions of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = e^{x+y}$ can be $e^x + e^{-y} = -2$.\nReason (R) : $e^x + e^{-y} = \text{C}$ is the general solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = e^{x+y}$.
(a) Find the general solution of the differential equation $(x^2 + y^2) \text{ d}y = xy \text{ d}x$. \
OR \
(b) Find the particular solution of the differential equation $\frac{\text{d}y}{\text{d}x} - 3y \cot x = \sin 2x$, given that $y = 2$ when $x = \frac{\pi}{2}$.
(a) Find the general solution of the differential equation
$(x^2 + y^2) \, dy = xy \, dx$.
OR
(b) Find the particular solution of the differential equation
$\frac{dy}{dx} - 3y \cot x = \sin 2x$, given that $y = 2$ when $x = \frac{\pi}{2}$.
$\frac{dy}{dx} = \text{F}(x, y)$ will be a homogeneous differential equation for which of the following functions ?
(i) $\text{F}(x, y) = 3x + 2y$
(ii) $\text{F}(x, y) = \sin \frac{y}{x} + \log y - \log x$
(iii) $\text{F}(x, y) = e^{y/x} + 1$
(iv) $\text{F}(x, y) = \sqrt{x^2 + y^2} - y$
(A) (i) and ...
Assertion (A) : One of the particular solutions of the differential equation $\frac{dy}{dx} = e^{x+y}$ can be $e^x + e^{-y} = -2$.
Reason (R) : $e^x + e^{-y} = C$ is the general solution of the differential equation $\frac{dy}{dx} = e^{x+y}$.
(a) Find the general solution of the differential equation $(x^{2} - y^{2}) dx + 2xy dy = 0$. OR (b) Solve the differential equation $\sin x \cos y dx + \cos x \sin y dy = 0$, given that $y = \frac{\pi}{4}$ when $x = 0$.
Product of the order and degree of differential equation $1 + \left(\frac{dy}{dx}\right)^3 = \lambda \left(\frac{d^3y}{dx^3}\right)^2$ is : (A) 5 (B) 6 (C) 2 (D) 3
Solve the differential equation $y e^y dx = (y^3 + 2x e^y) dy$, when $y(0) = 1$.
Find the general solution of the differential equation $(x^3 - 3xy^2) dx = (y^3 - 3x^2y) dy$.
The sum of the order and the degree of the differential equation $4 \left[ \frac{\text{d}^2y}{\text{dx}^2} \right]^2 + 3 \left[ 1 + \left$ \frac{\text{dy}}{\text{dx}} \right$^2 - y \right] = 0$ is : (A) $2$ (B) $3$ (C) not defined (D) $4$
Which of the following is not a Linear Differential Equation ?
(A) $(1 + x^2) dy + 2xy dx = \cot x dx$
(B) $y + \frac{d}{dx}(xy) = x (\sin x + \log x)$
(C) $x (1 + y^2) dx - y (1 + x^2) dy = 0$
(D) $y dx - (x + 3y^2) dy = 0$
The order and degree of the differential equation $1 + \left$ \frac{d^3y}{dx^3} \right$^3 = \lambda \frac{d^2y}{dx^2}$ is : (A) Order = 3, Degree = 3 (B) Order = 2, Degree = 2 (C) Order = 3, Degree = 1 (D) Order = 2, Degree = 1
Solve the differential equation $(x+2y^3) dy = y dx$.
The order and degree of differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2} = 1 - \left$ \frac{\text{d}^3\text{y}}{\text{dx}^3} \right$^2$ is :
(A) Order = 3, Degree = 3
(B) Order = 2, Degree = 2
(C) Order = 3, Degree = 2
(D) Order = 2, Degree = 1
Solve the differential equation $y \, dx + (x - y^3) \, dy = 0$.
Solve the differential equation $(x - \sin y) dy + \tan y dx = 0$.
The sum of the order and the degree of the differential equation : $\left(\frac{\text{d}y}{\text{d}x}\right)^5 + 3xy\left(\frac{\text{d}^3y}{\text{d}x^3}\right)^2 + y\left(\frac{\text{d}^2y}{\text{d}x^2}\right)^4 = 0$ is :
(A) 7
(B) 6
(C) 5
(D) 4
Solution of the differential equation $x \, \text{d}y + y \, \text{d}x = 0$ is :
(A) $x = Cy$
(B) $xy = C$
(C) $x^2y = C$
(D) $xy^2 = C$
The solution of the differential equation $\frac{\text{dy}}{\text{dx}} = e^{x-y}$ is :
(A) $e^y + e^x = C$
(B) $e^{-y} + e^x = C$
(C) $e^y - e^x = C$
(D) $e^x - e^{-y} = C$
The integrating factor of the differential equation $x \text{ dy} + (2y - x^3) \text{ dx} = 0$ is :
(A) $x$
(B) $\frac{1}{x}$
(C) $x^2$
(D) $\frac{1}{x^2}$
(a) Solve the differential equation $(3xy + y^2) \, \text{dx} + (x^2 + xy) \, \text{dy} = 0$. \
OR \
(b) Find the particular solution of the differential equation $x(1 + y^2) \, \text{dx} - y(1 + x^2) \, \text{dy} = 0$, given that $y(1) = 0$.
Solve the differential equation $x^2 \, dy + y(x + y) \, dx = 0$, given that $y = 1$ when $x = 1$.
Solve the differential equation $(x + y) \, dy + (x - y) \, dx = 0$, given that $y(1) = 1$.
Solve the differential equation $y + \frac{\text{d}}{\text{dx}}(xy) = x(\sin x + \log x)$.
The solution of the differential equation $xdx + ydy = 0$ represents a family of\n(A) straight lines\n(B) parabolas\n(C) Circles\n(D) Ellipses
Solve the differential equation: $y + \frac{d}{dx}(xy) = x(\sin x + x)$
Find the particular solution of the differential equation:\n$2 \, y \, e^{x/y} \, dx + (y - 2 \, x \, e^{x/y}) \, dy = 0$ given that $y(0) = 1$
(a) A given rectangular area is to be fenced off in a field whose length lies along a straight river. If no fencing is needed along the river, show that the least length of fencing will be required when the length of the rectangular area is twice its breadth.
OR
(b) Solve the differential equation...
The integrating factor for solving the differential equation $\tan x \frac{\text{dy}}{\text{dx}} + y = x^2, (x \neq 0)$ is :
(A) $e^x$
(B) $\tan x$
(C) $\sin x$
(D) $\frac{1}{\sin x}$
Find the particular solution of the differential equation $x \frac{dy}{dx} + y = x \cos x + \sin x$, given that $y = 1$ when $x = \frac{\pi}{2}$.
The general solution of the differential equation $\frac{\text{dy}}{\text{dx}} = 2x \cdot e^{x^2 + y}$ is :
(A) $e^{x^2 + y} = C$
(B) $e^{x^2} + e^{-y} = C$
(C) $e^{x^2} = e^y + C$
(D) $e^{x^2 - y} = C$
(a) Find the general solution of the differential equation $(2x^2 + y) \, dx = x \, dy$.\nOR\n(b) For the differential equation $\frac{dy}{dx} - \frac{y}{x} + \text{cosec}\left(\frac{y}{x}\right) = 0$, find the particular solution, given that $y = 0$ when $x = 1$.
The general solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = 2x . \mathrm{e}^{x^2 + y}$ is :
(A) $\mathrm{e}^{x^2 + y} = \mathrm{C}$
(B) $\mathrm{e}^{x^2} + \mathrm{e}^{-y} = \mathrm{C}$
(C) $\mathrm{e}^{x^2} = \mathrm{e}^y + \mathrm{C}$
(D) $\mathrm{e}^{x^2 - y} = \mathrm{C}...
If 'm' and 'n' are the degree and order respectively of the differential equation $1 + \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^3 = \frac{\mathrm{d}^2y}{\mathrm{dx}^2}$, then the value of $(m + n)$ is :
(A) $4$
(B) $3$
(C) $2$
(D) $5$
(a) Find the general solution of the differential equation $x^2 \frac{\text{dy}}{\text{dx}} = x^2 + xy + y^2$.
OR
(b) Find the particular solution of the differential equation $\frac{\text{dy}}{\text{dx}} - 3y \cot x = \sin 2x$, given that $y = 2$ when $x = \frac{\pi}{2}$.
The integrating factor of the differential equation $\frac{\text{dy}}{\text{dx}} + \text{y} = \frac{1 + \text{y}}{\text{x}}$ is :
(A) $\log \text{x}$
(B) $-\log \text{x}$
(C) $\text{e}^{\text{x}} - \text{x}$
(D) $\frac{\text{e}^{\text{x}}}{\text{x}}$
(a) Find the general solution of the differential equation
$(2x^2 + y) \text{dx} = x \text{dy}$.
OR
(b) For the differential equation $\frac{\text{dy}}{\text{dx}} - \frac{y}{x} + \text{cosec}\left(\frac{y}{x}\right) = 0$, find the particular solution, given that $y = 0$ when $x = 1$.
The order and degree of the differential equation $\left[1 + \left(\frac{\text{dy}}{\text{dx}}\right)^2\right]^3 = \frac{\text{d}^2\text{y}}{\text{dx}^2}$ respectively are :
(A) $1, 2$
(B) $2, 3$
(C) $2, 1$
(D) $2, 6$
Find the particular solution of the differential equation given by $x^2 \frac{dy}{dx} - xy = x^2 \cos^2 \left(\frac{y}{2x}\right)$, given that when $x = 1, y = \frac{\pi}{2}$.
The integrating factor of the differential equation $x \frac{dy}{dx} - y = x^4 - 3x$ is :
(A) $x$
(B) $-x$
(C) $x^{-1}$
(D) $\log(x^{-1})$
The integrating factor of differential equation $(x + 2y^3) \frac{dy}{dx} = 2y$ is
(A) $e^{\frac{y^2}{2}}$
(B) $\frac{1}{\sqrt{y}}$
(C) $\frac{1}{y^2}$
(D) $e^{-\frac{1}{y^2}}$
The integrating factor of the differential equation $\frac{\text{dy}}{\text{dx}} + y = \frac{1 + y}{x}$ is
(A) $x e^x$
(B) $\frac{e^x}{x}$
(C) $\frac{x}{e^x}$
(D) $x e^{\frac{1}{x}}$
If $p$ and $q$ are respectively the order and degree of the differential equation $\frac{\text{d}}{\text{d}x} \left(\frac{\text{d}y}{\text{d}x}\right)^3 = 0$, then $(p - q)$ is
(A) $0$
(B) $1$
(C) $2$
(D) $3$
If p and q are respectively the order and degree of the differential equation $\frac{d}{dx}\left(\frac{dy}{dx}\right)^3 = 0$, then (p - q) is
(A) 0
(B) 1
(C) 2
(D) 3
(a) Solve the differential equation $2(y + 3) - xy \frac{\text{d}y}{\text{d}x} = 0$; given $y(1) = -2$.
OR
(b) Solve the following differential equation :
$(1 + x^2) \frac{\text{d}y}{\text{d}x} + 2xy = 4x^2$
The integrating factor for solving the differential equation $x \cdot \frac{dy}{dx} - y = 2x^2$ is
(A) $x$
(B) $\frac{1}{x}$
(C) $e^{-x}$
(D) $-\log x$
Assertion (A) : $x^2 \text{dy} = (2xy + y^2)\text{dx}$ is a homogeneous differential equation.\nReason (R) : A differential equation of the form $\frac{\text{dy}}{\text{dx}} = F\left(\frac{y}{x}\right)$ is a homogeneous differential equation.
(a) Find the particular solution of the differential equation, $\frac{dy}{dx} = 1 + x^2 + y^2 + x^2y^2$, given that $y = 1$ when $x = 0$.
OR
(b) Solve the differential equation : $2xy \frac{dy}{dx} = x^2 + 3y^2$.
The integrating factor of the differential equation
$\frac{dy}{dx} + y \tan x - \sec x = 0$ is :
(A) $-\cos x$
(B) $\sec x$
(C) $\log \sec x$
(D) $e^{\sec x}$
The integrating factor of the differential equation $\left$\frac{e^{-2\sqrt{x}}}{\sqrt{x}} - \frac{y}{\sqrt{x}}\right$ \frac{dx}{dy} = 1$ is : (A) $e^{-1/\sqrt{x}}$ (B) $e^{2/\sqrt{x}}$ (C) $e^{2\sqrt{x}}$ (D) $e^{-2\sqrt{x}}$
Find the particular solution of the differential equation \
\[\left[x \sin^2\left(\frac{y}{x}\right) - y\right] dx + x \, dy = 0\] \
given that $y = \frac{\pi}{4}$, when $x = 1$.
The integrating factor of the differential equation $\frac{dx}{dy} = \frac{x \log x}{\frac{2}{x} \log x - y}$ is :
(A) $\frac{1}{8x}$
(B) $e$
(C) $e^{\log x}$
(D) $\log x$
The sum of the order and degree of the differential equation $\left[1 + \left(\frac{dy}{dx}
ight)^2
ight]^3 = \frac{d^2y}{dx^2}$ is :
(A) $2$
(B) $\frac{5}{2}$
(C) $3$
(D) $4$
Find the particular solution of the differential equation $\frac{\text{d}y}{\text{d}x} - \frac{y}{x} + \text{cosec}\left(\frac{y}{x}\right) = 0$; given that $y = 0$, when $x = 1$.
The integrating factor of the differential equation $\frac{dx}{dy} = \frac{-(1 + \sin x)}{x + y \cos x}$ is : (A) log cos x (B) 1 + sin x (C) e$^{(1 + \sin x)}$ (D) e$^{\log \cos x}$
The order and degree of the following differential equation are, respectively : $-\frac{d^4y}{dx^4} + 2e^{dy/dx} + y^2 = 0$. (A) $-4, 1$ (B) $4, \text{not defined}$ (C) $1, 1$ (D) $4, 1$
(a) Solve the differential equation : $x^2 y\, dx - (x^3 + y^3)\, dy = 0$. \
OR \
(b) Solve the differential equation $(1 + x^2) \frac{dy}{dx} + 2xy - 4x^2 = 0$ subject to initial condition $y(0) = 0$.
The order and degree of the differential equation $\left[\frac{\text{d}^2y}{\text{d}x^2} - 1\right]^2 = \frac{\text{d}y}{\text{d}x}$ are,
respectively :
(A) $2, 2$
(B) $2, \text{not defined}$
(C) $1, 2$
(D) $1, \text{not defined}$
Solve the differential equation $(x^2 + y^2) dx + xy dy = 0, y(1) = 1$.
Solve the differential equation $(x - \sin y) dy + (\tan y) dx = 0$, given $y(0) = 0$.
Q.14. What is the general solution of the differential equation $e^{y'} = x$?
(A) $y = x\log x + c$
(B) $y = x\log x - x + c$
(C) $y = x\log x + x + c$
(D) $y = x + c$
The differential equation $\frac{\text{dy}}{\text{dx}} = F(x, y)$ will not be a homogeneous differential equation, if $F(x, y)$ is :
(A) $\cos x - \sin \left$ \frac{y}{x} \right$$
(B) $\frac{y}{x}$
(C) $\frac{x^2 + y^2}{\text{xy}}$
(D) $\cos^2 \left$ \frac{x}{y} \right$$
The degree of the differential equation (y'')$^2$ + (y')$^3$ = x sin (y') is : (A) 1 (B) 2 (C) 3 (D) not defined
The number of solutions of differential equation $\frac{\text{dy}}{\text{dx}} - y = 1$, given that $y(0) = 1$, is :
(A) 0
(B) 1
(C) 2
(D) infinitely many
The degree of the differential equation $(y'')^2 + (y')^3 = x \sin(y')$ is :
(A) $1$
(B) $2$
(C) $3$
(D) not defined
Find the particular solution of the differential equation given by $2xy + y^2 - 2x^2 \frac{dy}{dx} = 0$; $y = 2$, when $x = 1$.
Find the general solution of the differential equation : $y dx = (x + 2y^2) dy$
The degree and order of differential equation $y''^2 + \log(y') = x^5$ respectively are :
(A) not defined, 5
(B) not defined, 2
(C) 5, not defined
(D) 2, 2
The differential equation $\frac{dy}{dx} = F(x, y)$ will not be a homogeneous differential equation, if $F(x, y)$ is :
(A) $\cos x - \sin(\frac{y}{x})$
(B) $\frac{y}{x}$
(C) $\frac{x^2 + y^2}{xy}$
(D) $\cos^2(\frac{x}{y})$
The integrating factor of the differential equation $\frac{dy}{dx} + y \tan x = 2x + x^2 \tan x$ is :
(A) $e^{\sec x}$
(B) $\sec x + \tan x$
(C) $\sec x$
(D) $\cos x$
(a) Find the particular solution of the differential equation $2xy \frac{dy}{dx} = x^2 + 3y^2$, given that $y(1) = 0$. OR (b) Solve the differential equation $\frac{dy}{dx} + 2y \tan x = \sin x$, given that $y = 0$, when $x = \frac{\pi}{3}$.
$x \log x \frac{\text{dy}}{\text{dx}} + y = 2 \log x$ is an example of a :
(A) variable separable differential equation.
(B) homogeneous differential equation.
(C) first order linear differential equation.
(D) differential equation whose degree is not defined.
The general solution of the differential equation $x \, dy + y \, dx = 0$ is :
(A) $xy = c$
(B) $x + y = c$
(C) $x^2 + y^2 = c^2$
(D) $\log y = \log x + c$
(a) Find the particular solution of the differential equation $\frac{dy}{dx} = y \cot 2x$, given that $y$\frac{\pi}{4}$ = 2$. OR (b) Find the particular solution of the differential equation $$xe^{\frac{y}{x}} + y$ dx = x dy$, given that $y = 1$ when $x = 1$.
The number of arbitrary constants in the particular solution of the differential equation \log \left$ \frac{dy}{dx} \right$ = 3x + 4y; y(0) = 0 is/are (A) 2 (B) 1 (C) 0 (D) 3
Find the particular solution of the differential equation $\frac{dy}{dx} = y \cot 2x$, given that $y\left$\frac{\pi}{4}\right$ = 2$.
Find the particular solution of the differential equation $$x e^{\frac{y}{x}} + y$ dx = x dy$, given that $y = 1$ when $x = 1$.
The integrating factor of the differential equation $\frac{dy}{dx} + \frac{2}{x}y = 0, x \neq 0$ is :
(A) $\frac{2}{x}$
(B) $x^{2}$
(C) $e^{\frac{2}{x}}$
(D) $e^{\log(2x)}$
(a) Find the particular solution of the differential equation $\frac{dy}{dx} = y \cot 2x$, given that $y\left$\frac{\pi}{4}\right$ = 2$. OR (b) Find the particular solution of the differential equation $$x e^{\frac{y}{x}} + y$ \, dx = x \, dy$, given that $y = 1$ when $x = 1$.
The general solution of the differential equation $\frac{dy}{dx}=e^{x+y}$ is :
(A) $e^{x}+e^{-y}=c$
(B) $e^{-x}+e^{-y}=c$
(C) $e^{x+y}=c$
(D) $2e^{x+y}=c$
The general solution of the differential equation
$\frac{\text{dy}}{\text{dx}}=\text{e}^{x+y}$ is :
(A) $\text{e}^{x}+\text{e}^{-y}=\text{c}$
(B) $\text{e}^{-x}+\text{e}^{-y}=\text{c}$
(C) $\text{e}^{x}+\text{y}=\text{c}$
(D) $2\text{e}^{x}+\text{y}=\text{c}$
Find the particular solution of the differential equation $\frac{\text{dy}}{\text{dx}} - 2xy = 3x^{2} e^{x^{2}}$ ; $y(0) = 5$.
Solve the following differential equation : $x^{2} \text{dy} + y(x + y) \text{dx} = 0$
Find the particular solution of the differential equation $\frac{dy}{dx} - 2xy = 3x^2 e^{x^2} ; y(0) = 5$.
Solve the following differential equation : $x^2 dy + y(x + y) dx = 0$
The number of arbitrary constants in the general solution of the differential equation $\frac{dy}{dx}+y=0$ is:
Find the general solution of the differential equation $y dx - x dy + (x \log x) dx = 0$.
The integrating factor of the differential equation $(x^2 - 1) \frac{dy}{dx} + 2xy = \frac{1}{x^2 - 1}$ is :
(A) $e^{2x}$
(B) $e^{x^2 - 1}$
(C) $\log (x^2 - 1)$
(D) $x^2 - 1$
Find the general solution of the differential equation $\frac{dy}{dx} = xy \log x \log y$.
Find the particular solution of the differential equation $\frac{dy}{dx} + \sec^{2} x \cdot y = an x \cdot \sec^{2} x$, given that $y(0) = 0$.
Solve the differential equation given by $x dy - y dx - \sqrt{x^{2} + y^{2}} dx = 0$.
Find the general solution of the differential equation: $\frac{dx}{dy} = \frac{e^{x/y} \left$ \frac{x}{y} - 1 \right$}{1 + e^{x/y}}$
Find the particular solution of the differential equation $\frac{dy}{dx} + \cot x \cdot y = \cos^2 x$, given that when $x = \frac{\pi}{2}, y = 0$.
The general solution of the differential equation x dy - (1 + x^2) dx = dx is :
(a) y = 2x + \frac{x^3}{3} + C
(b) y = 2 \log x + \frac{x^3}{3} + C
(c) y = \frac{x^2}{2} + C
(d) y = 2 \log x + \frac{x^2}{2} + C
Find the general solution of the differential equation : $\frac{\text{dy}}{\text{dx}} - \frac{2y}{x} = \sin \frac{1}{x}$.
Find the particular solution of the differential equation : $\frac{\text{dy}}{\text{dx}} = \sin(x + y) + \sin(x - y)$, given that when $x = \frac{\pi}{4}, y = 0$.
The general solution of the differential equation $y dx - x dy = 0$; (Given $x, y > 0$), is of the form
(a) $xy = c$
(b) $x = cy^2$
(c) $y = cx$
(d) $y = cx^2$;
(Where 'c' is an arbitrary positive constant of integration)
Solve the differential equation: $ye^{\frac{x}{y}} dx = \left$ xe^{\frac{x}{y}} + y^{2} \right$ dy, (y \neq 0)$. OR Solve the differential equation: $(\cos^{2} x) \frac{dy}{dx} + y = \tan x; \left( 0 \leq x < \frac{\pi}{2} \right)$.
Find the general solution of the differential equation : $(xy - x^2) dy = y^2 dx$.
Find the general solution of the differential equation : $(x^2 + 1) \frac{dy}{dx} + 2xy = \sqrt{x^2 + 4}$
The general solution of the differential equation $\frac{dy}{dx} = e^{x-y}$ is : (a) $e^x + e^y = C$ (b) $e^x - e^{-y} = C$ (c) $-e^x - e^y = C$ (d) $e^x - e^y = C$
(a) Find a particular solution of the differential equation $\frac{\text{dy}}{\text{dx}} + \text{y} \cot \text{x} = 4\text{x} \operatorname{cosec} \text{x}$, $(\text{x} \neq 0)$, given that $\text{y} = 0$ when $\text{x} = \frac{\pi}{2}$.
(b) Solve the differential equation $\text{x} \text{dy} - \text{y} \text{dx} = \sqrt{\text{x}^{2} + \text{y}^{2}} \text{dx}$.
The solution of the differential equation $\frac{dx}{x} + \frac{dy}{y} = 0$ is :
(a) $\frac{1}{x} + \frac{1}{y} = C$
(b) $\log x - \log y = C$
(c) $xy = C$
(d) $x + y = C$
The solution of the differential equation $\frac{\text{dx}}{\text{x}} + \frac{\text{dy}}{\text{y}} = 0$ is :
(a) $\frac{1}{\text{x}} + \frac{1}{\text{y}} = \text{C}$
(b) $\log \text{x} - \log \text{y} = \text{C}$
(c) $\text{xy} = \text{C}$
(d) $\text{x} + \text{y} = \text{C}$
What is the product of the order and degree of the differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2} \sin \text{y} + \left(\frac{\text{dy}}{\text{dx}}\right)^3 \cos \text{y} = \sqrt{\text{y}}$ ?
(a) 3
(b) 2
(c) 6
(d) not defined
The integrating factor for solving the differential equation $x \frac{\text{dy}}{\text{dx}} - y = 2x^2$ is :
(a) $e^{-y}$
(b) $e^{-x}$
(c) $x$
(d) $\frac{1}{x}$
(a) Find the particular solution of the differential equation
$$\frac{dy}{dx} = \frac{x + y}{x}, \quad y(1) = 0.$$
OR
(b) Find the general solution of the differential equation
$$e^x \tan y \, dx + (1 - e^x) \sec^2 y \, dy = 0.$$
Case Study-III
An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form $\frac{dy}{dx} = F(x, y)$ is said to be homogeneous if $F(x, y)$ is a homogeneous function of degree zero, where...
Case Study-III
38. An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form $\frac{dy}{dx} = F(x, y)$ is said to be homogeneous if $F(x, y)$ is a homogeneous function of degree zero, ...
The order of the differential equation corresponding to family of curves y = Ae^{3x} + Be^{-3x} is :
(a) 1
(b) 2
(c) 3
(d) 4
The difference of the order and the degree of the differential equation $\left(\frac{\text{d}^2y}{\text{d}x^2}\right)^2 + \left(\frac{\text{d}y}{\text{d}x}\right)^3 + x^4 = 0$ is :
(a) $1$
(b) $2$
(c) $-1$
(d) $0$
The integrating factor of the differential equation $(3x^2 + y)\frac{\text{d}x}{\text{d}y} = x$ is
(a) $\frac{1}{x}$
(b) $\frac{1}{x^2}$
(c) $\frac{2}{x}$
(d) $-\frac{1}{x}$
(a) Find the particular solution of the differential equation $\frac{\text{dy}}{\text{dx}} = \frac{\text{xy}}{\text{x}^2 + \text{y}^2}$, given that y = 1 when x = 0. OR (b) Find the particular solution of the differential equation $(1 + \text{x}^2)\frac{\text{dy}}{\text{dx}} + 2\text{xy} = \frac{1}{...
(a) Find the particular solution of the differential equation $\frac{dy}{dx} = \frac{xy}{x^2 + y^2}$, given that y = 1 when x = 0. OR (b) Find the particular solution of the differential equation $(1 + x^2)\frac{dy}{dx} + 2xy = \frac{1}{1 + x^2}$, given that y = 0 when x = 1.
Integrating factor of the differential equation $x \frac{dy}{dx} - 2y = 4x^2$ is : (a) $x^2$ (b) $-\frac{1}{x^2}$ (c) $\frac{1}{x^2}$ (d) $-x^2$
The difference of the order and the degree of the differential equation
$(\frac{\text{d}^2y}{\text{dx}^2})^2 + (\frac{\text{dy}}{\text{dx}})^3 + x^4 = 0$ is:
(a) $1$
(b) $2$
(c) $-1$
(d) $0$
Find the particular solution of the differential equation $\frac{\text{dy}}{\text{dx}} = \frac{xy}{x^2 + y^2}$, given that $\text{y} = 1$ when $\text{x} = 0$.
Find the particular solution of the differential equation $(1 + x^2) \frac{\text{dy}}{\text{dx}} + 2xy = \frac{1}{1 + x^2}$, given that $\text{y} = 0$ when $\text{x} = 1$.
The sum of the order and the degree of the differential equation $3x^{2} \left$ \frac{d^{2}y}{dx^{2}} \right$^{3} + 5 \frac{d^{3}y}{dx^{3}} + 9 = 0$ is: (a) 5 (b) 6 (c) 4 (d) 2
A particular solution of the differential equation $dy = y \tan x \, dx$; $y = 1$ when $x = 0$, is: (a) $y = 0$ (b) $y = 1 + \sec x$ (c) $y = \sec x$ (d) $y \cos x = 0$
(a) Solve the differential equation $x \frac{dy}{dx} = y - x \tan \frac{y}{x}$.
OR
(b) Find the particular solution of the differential equation $x \frac{dy}{dx} + y = x \cos x + \sin x$; given $y \left$ \frac{\pi}{2} \right$ = 1$.
Find the particular solution of the differential equation $x \frac{\text{d}y}{\text{d}x} + x \cos^2\left(\frac{y}{x}\right) = y$; given that when $x = 1$, $y = \frac{\pi}{4}$.
Find the particular solution of the differential equation $x \frac{dy}{dx} + x \cos^2(\frac{y}{x}) = y$; given that when $x = 1$, $y = \frac{\pi}{4}$.
Find the product of the order and the degree of the differential equation $\left[\frac{\text{d}}{\text{dx}}(xy^{2})\right] \cdot \frac{\text{dy}}{\text{dx}} + y = 0$.
Find the particular solution of the differential equation $(y + 3x^2) \frac{dx}{dy} = x$, given that y = 1, when x = 1.
Find the particular solution of the differential equation $\frac{dy}{dx} = 2y^2$, given y = 1 when x = 1.
Find the general solution of the differential equation x $\frac{dy}{dx}$ - y + x sin $\frac{y}{x}$ = 0.
The general solution of the differential equation $ydx - xdy = 0$ is
(a) $xy = C$ (b) $x = Cy^2$ (c) $y = Cx$ (d) $y = Cx^2$
Solve the differential equation: $ydx + (x - y^2)dy = 0$
OR
Solve the differential equation: $xdy - ydx = \sqrt{x^2 + y^2} \, dx$
Solve the following differential equation : $(1 + e^{y/x}) \text{dy} + e^{y/x} \left$ 1 - \frac{y}{x} \right$ \text{dx} = 0$
Find the general solution of the differential equation : $e^{dy/dx} = x^2$.
Find the general solution of the differential equation : $\log \left$ \frac{dy}{dx} \right$ = ax + by$.
Find the general solution of the differential equation : $\frac{ ext{dy}}{ ext{dx}} = \frac{3 ext{e}^{2x} + 3 ext{e}^{4x}}{ ext{e}^x + ext{e}^{-x}}$
Find the particular solution of the differential equation $x \frac{\text{d}y}{\text{d}x} - y = x^2 \cdot e^x$, given $y(1) = 0$.
OR
Find the general solution of the differential equation $x \frac{\text{d}y}{\text{d}x} = y(\log y - \log x + 1)$.
Find the general solution of the differential equation $\sec^2 x \cdot an y \, dx + \sec^2 y \cdot an x \, dy = 0$.
(a) Find the particular solution of the differential equation $x \frac{dy}{dx} + y + \frac{1}{1 + x^2} = 0$, given that $y(1) = 0$. OR (b) Find the general solution of the differential equation $x (y^3 + x^3) dy = (2y^4 + 5x^3y) dx$.
Solve the following differential equation : $(y - \sin^2 x) dx + \tan x dy = 0$
Find the general solution of the differential equation : $(x^3 + y^3) dy = x^2 y dx$
(a) Find the general solution of the differential equation x cos y dy = (x log x + 1) e^x dx.
Find the particular solution of the differential equation $x \frac{dy}{dx} + 2y = x^2 \log x$, given $y(1) = 1$.
(a) Find the general solution of the differential equation $x \cos y dy = (x \log x + 1) e^x dx$.
Find the particular solution of the differential equation $(1 + \sin x) \frac{dy}{dx} = -x - y \cos x$, given $y(0) = 1$.
(a) Find the general solution of the differential equation $x \cos y \, dy = (x \log x + 1) e^x \, dx$.
Find the particular solution of the differential equation $(1 + x^2) \frac{dy}{dx} + 2xy = \tan x$, given $y(0) = 1$.
Find the general solution of the differential equation : $\frac{dy}{dx} = (1 + x^2)(1 + y^2)$
Find the particular solution of the differential equation : $(x \cos^2(y/x) - y) \text{d}x + x \text{d}y = 0$, given that $y = \frac{\pi}{4}$ when $x = 1$.
Find the general solution of the differential equation : $(x + y) \frac{\text{d}x}{\text{d}y} = 1$