Differential Equations — JEE Main PYQs

If for $x\ge 0$, $y=y(x)$ is the solution of the differential equation, $(x+1)dy=\left((x+1{)}^2+y-3\right)dx$, $y(2)=0$, then $y(3)$ is equal to ______

Let $y=y(x)$ be the solution of the differential equation $\cos x\frac{dy}{dx}+2y\sin x=\sin 2x,x\in \left(0,\frac{\pi }{2}\right)$.
If $y(\pi /3)=0$, then $y(\pi /4)$ is equal to:

The solution curve of the differential equation, $\left(1+e^{-x}\right)\left(1+y^2\right)\frac{dy}{dx}=y^2$, which passes through the point ( 0,1 ), is

If $x^{3}dy+xydx=x^{2}dy+2ydx;y(2)=e$ and $x>1$, then $y(4)$ is equal to:

If $y=\left(\frac{2}{\pi }x-1\right)\mathrm{cosec}x$ is the solution of the differential equation, $\frac{dy}{dx}+p(x)y=\frac{2}{\pi }\mathrm{cosec}x,0<x<\frac{\pi }{2}$, then the function $p(x)$ is equal to:

If $y=y(x)$ is the solution of the differential equation $\frac{5+e^x}{2+y}\cdot \frac{dy}{dx}+e^x=0$ satisfying $y(0)=1$, then a value of $y\left({\log }_{e}13\right)$ is:

The general solution of the differential equation $\sqrt{1+x^2+y^2+x^2{y}^2}+xy\frac{dy}{dx}=0$ is: (where C is a constant of integration)

Let $y=y(x)$ be the solution of the differential equation, $x{y}^{\prime }-y=x^2(x\cos x+\sin x),x>0$. If $y(\pi )=\pi$, then $y^{\prime \prime }\left(\frac{\pi }{2}\right)+y\left(\frac{\pi }{2}\right)$ is equal to:

The solution of differential equation $\frac{dy}{dx}-\frac{y+3x}{{\log }_e(y+3x)}+3=0$ is: (where C is a constant of integration.)

Let $y=y(x)$ be the solutions of the differential equation, ${\left(x^2+1\right)}^2\frac{dy}{dx}+2x\left(x^2+1\right)y=1$ such that $y(0)=0$. If $\sqrt{a}y(1)=\frac{\pi }{32}$, then the value of '$a$' is

Given that the slope of the tangent to a curve $y=y(x)$ at any point $(x,y)$ is $\frac{2y}{x^2}$. If the curve passes through the centre of the circle $x^2+y^2-2x-2y=0$, then its equation is:

The solution of the differential equation $x\frac{dy}{dx}+2y=x^2(x\ne 0)$ with $y(1)=1$, is:

If $\frac{dy}{dx}+\frac{3}{{\cos }^{2}x}y=\frac{1}{{\cos }^{2}x},x\in \left(\frac{-\pi }{3},\frac{\pi }{3}\right)$ and $y\left(\frac{\pi }{4}\right)=\frac{4}{3}$, then $y\left(-\frac{\pi }{4}\right)$ equals:

The solution of the differential equation $\frac{dy}{dx}=(x-y{)}^2$ when $y(1)=1$, is:

Consider the differential equation, $y^{2}dx+\left(x-\frac{1}{y}\right)dy=0$. If value of $y$ is 1 when $x=1$, then the value of x for which $\mathrm{y}=2$, is:

Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $\frac{dy}{dx}+y\tan x=2x+x^2\tan x,x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, such that $y(0)=1$. Then

If $y(x)$ is the solution of the differential equation $\frac{dy}{dx}+\left(\frac{2x+1}{x}\right)y=e^{-2x},x>0$ where $y(1)=\frac{1}{2}{\mathrm{e}}^{-2}$, then:

Let $y=y(x)$ be the solution of the differential equation, $x\frac{dy}{dx}+y=x{\log }_{e}x,(x>1)$. If $2y(2)={\log }_{\mathrm{e}}4-1$, then $\mathrm{y}(\mathrm{e})$ is equal to:

The general solution of the differential equation $(y^2-x^3)dx-xydy=0$ $(x\ne 0)$ is : (where c is a constant of integration)

If a curve passes through the point ( $1,-2$ ) and has slope of the tangent at any point ( x , $y)$ on it as $\frac{x^2-2y}{x}$, then the curve also passes through the point :

If $\cos x\frac{dy}{dx}-y\sin x=6x,\left(0<x<\frac{\pi }{2}\right)$ and $y\left(\frac{\pi }{3}\right)=0$, then $y\left(\frac{\pi }{6}\right)$ is equal to:

Let $f:[0,1]\to R$ be such that $f(xy)=f(x)f(y)$ for all $x,y\in [0,1]$, and $f(0)\ne 0$. If $y=y(x)$ satisfies the differential equation, $\frac{dy}{dx}=f(x)$ with $y(0)=1$, then $y\left(\frac{1}{4}\right)+y\left(\frac{3}{4}\right)$ is equal to

If $y=y(x)$ is solution of the differential equation $x\frac{dy}{dx}+2y=x^2$ satisfying $y(1)=1$, then $y\left(\frac{1}{2}\right)$ is equal to

If $y=f(x)$ is the solution of the differential equation $\frac{dy}{dx}=(\tan x-y){\sec }^{2}x$, $x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, such that $y(0)=0$, then $y\left(-\frac{\pi }{4}\right)$ is equal to

Let $y=y(x)$ be the solution of the differential equation
$\sin x\frac{\mathrm{d}y}{\mathrm{d}x}+y\cos x=4x,x\in (0,\pi )$. If
$y\left(\frac{\pi }{2}\right)=0$, then $y\left(\frac{\pi }{6}\right)$ is equal to :

If $(2+\sin x)\frac{\mathrm{d}y}{\mathrm{d}x}+(y+1)\cos x=0$ and $y(0)=1$, then $y\left(\frac{\pi }{2}\right)$ is equal to :

If a curve $y=f(x)$ passes through the point (1,-1) and satisfies the differential equation, $y(1+xy)dx=xdy$, then $f\left(-\frac{1}{2}\right)$ is equal to:

Let $\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $(\mathrm{x}\log \mathrm{x})\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}=2\mathrm{x}\log \mathrm{x},(\mathrm{x}\ge 1)$. Then $\mathrm{y}(\mathrm{e})$ is equal to :

Let the population of rabbits surviving at a time $t$ be governed by the differential equation $\frac{dp(t)}{dt}=\frac{1}{2}p(t)-200$. If $\mathrm{p}(0)=100$, then $p(t)$ equals

The population $p(t)$ at time $t$ of a certain mouse species satisfies the differential equation $\frac{dp(t)}{dt}=0.5p(t)$ -450 . If $p(0)=850$, then the time at which the population becomes zero is

If $\frac{dy}{dx}=y+3>0$ and $y(0)=2$, then $y(\ln 2)$ is equal to

Let I be the purchase value of an equipment and $V(t)$ be the value after it has been used for $t$ years. The value $V(t)$ depreciates at a rate given by differential equation $\frac{dV(t)}{dt}=-k(T-t)$, where $k>0$ is a constant and $T$ is the total life in years o …

Solution of the differential equation $\cos xdy=y(\sin x-y)dx,0<x<\frac{\pi }{2}$ is

The differential equation which represents the family of curves $y=c_1{e}^{c_{2}x}$, where $c_1$ and $c_2$ are arbitrary constants is

The solution of the differential equation $\frac{dy}{dx}=\frac{x+y}{x}$ satisfying the condition $y(1)=1$ is

The differential equation of the family of circles with fixed radius 5 units and centre on the line $y=2$ is

The differential equation of all circles passing through the origin and having their centres on the x-axis is

The normal to a curve at $P(x,y)$ meets the $x$-axis at $G$. If the distance of $G$ from the origin is twice the abscissa of $P$, then the curve is a

The differential equation whose solution is $A{x}^2+B{y}^2=1$, where $A$ and $B$ are arbitrary constants is of

The differential equation representing the family of curves $y^2=2c(x+\sqrt{c})$, where $c>0$, is a parameter, is of order and degree as follows:

If $x\frac{dy}{dx}=y(\log y-\log x+1)$, then the solution of the equation is

The differential equation for the family of curves $x^2+y^2-2ay=0$, where $a$ is an arbitrary constant is

The solution of the differential equation $ydx+\left(x+x^{2}y\right)dy=0$ is