Differential Equations — Class 12 Mathematics PYQs

Which of the following is not a homogeneous function of $x$ and $y$ ?

(A) $y^2-xy$
(B) $x-3y$
(C) ${\sin }^2\left(\frac{y}{x}\right)+\frac{y}{x}$
(D) $\tan x-\sec y$

The integrating factor of differential equation $(x+2y^3)\frac{dy}{dx}=2y$ is

(A) $e^{y^2/2}$
(B) $\frac{1}{\sqrt{y}}$
(C) $\frac{1}{y^2}$
(D) $e^{-1/y^2}$

The integrating factor of the differential equation $\frac{dy}{dx}+y=\frac{1+y}{x}$ is

(A) $x{e}^x$
(B) $\frac{e^x}{x}$
(C) $\frac{x}{e^x}$
(D) $x{e}^{1/x}$

The order and degree of differential equation ${\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^5=\frac{d^{2}y}{d{x}^2}$ are

(A) order 1, degree 1
(B) order 1, degree 2
(C) order 2, degree 1
(D) order 2, degree 2

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 $2(y+3)-xy\frac{dy}{dx}=0$; given $y(1)=-2$.

OR

(b) Solve the following differential equation :

$$(1+x^2)\frac{dy}{dx}+2xy=4x^2$$

(a) If $y=\log {\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)}^2$, then show that $x(x+1{)}^2{y}_2+(x+1{)}^2{y}_1=2$.

OR

(b) If $x\sqrt{1+y}+y\sqrt{1+x}=0$, $-1<x<1$, $x\ne y$, then prove that $\frac{dy}{dx}=\frac{-1}{(1+x{)}^2}$.

The number of solutions of differential equation $\frac{dy}{dx}-y=1$, given that $y(0)=1$, is :

The degree and order of differential equation $(y^{\prime \prime }{)}^2+\log (y^{\prime })=x^5$ respectively are :

The integrating factor of the differential equation $x\frac{dy}{dx}-y=x^4-3x$ is :