Differential Equations — JEE Main PYQs

Let f be a differentiable function defined on $\left[0,\frac{\pi }{2}\right]$ such that $f(x)>0$ and $f(x)+{\int }_0^{x}f(t)\sqrt{1-({\log }_{e}f(t){)}^2}dt=e,\forall x\in \left[0,\frac{\pi }{2}\right]$. Then ${\left(6{\log }_{e}f\left(\frac{\pi }{6}\right)\right)}^2$ is equal to

Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation
$\left(3y^2-5x^2\right)ydx+2x\left(x^2-y^2\right)dy=0$ such that $\mathrm{y}(1)=1$. then $\left|(\mathrm{y}(2){)}^3-12\mathrm{y}(2)\right|$ is equal to:

The solution of the differential equation
$\frac{\mathrm{dy}}{\mathrm{dx}}=-\left(\frac{{\mathrm{x}}^2+3{\mathrm{y}}^2}{3{\mathrm{x}}^2+{\mathrm{y}}^2}\right),\mathrm{y}(1)=0$ is

Let a differentiable function f satisfy $f(x)+{\int }_3^x\frac{f(t)}{t}dt=\sqrt{x+1},x\ge 3$. Then 12f(8) is equal to:

If $y=y(x)$ is the solution curve of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}\tan \mathrm{x}=\mathrm{x}\sec \mathrm{x},0\le \mathrm{x}\le \frac{\pi }{3}$, $y(0)=1$, then $y\left(\frac{\pi }{6}\right)$ is equal to

The area enclosed by the closed curve C given by the differential equation $\frac{dy}{dx}+\frac{x+a}{y-2}=0,y(1)=0$ is $4\pi$.

Let P and Q be the points of intersection of the curve C and the y -axis. If normals at P and Q on the curve C intersect x -axis at points R and …

Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function such that $f^{\prime }(x)+f(x)={\int }_0^{2}f(t)dt$. If $f(0)=e^{-2}$, then $2f(0)-f(2)$ is equal to ____.

Let $y=y(x)$ be the solution curve of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}}{\mathrm{x}}\left(1+{\mathrm{xy}}^2\left(1+{\log }_{\mathrm{e}}\mathrm{x}\right)\right)$, $x>0,y(1)=3$. Then $\frac{y^2(x)}{9}$ is equal to :

If the solution curve of the differential equation $\frac{dy}{dx}=\frac{x+y-2}{x-y}$ passes through the point $(2,1)$ and $(\mathrm{k}+1,2),\mathrm{k}>0$, then

Let $y=y(x)$ be the solution curve of the differential equation $\frac{dy}{dx}+\left(\frac{2x^2+11x+13}{x^3+6x^2+11x+6}\right)\mathrm{y}=\frac{(\mathrm{x}+3)}{\mathrm{x}+1},\mathrm{x}>-1$, which passes through the point $(0,1)$. Then $\mathrm{y}(1)$ is equal to:

Let the solution curve of the differential equation $xdy=\left(\sqrt{x^2+y^2}+y\right)dx,x>0$, intersect the line $\mathrm{x}=1$ at $\mathrm{y}=0$ and the line $\mathrm{x}=2$ at $\mathrm{y}=\alpha$. Then the value of $\alpha$ is :

If $y=y(x),x\in \left(0,\frac{\pi }{2}\right)$ be the solution curve of the differential equation $2{\mathrm{e}}^{-4\mathrm{x}}(2\sin 2\mathrm{x}+\cos 2\mathrm{x}),$ with …

The minimum value of the twice differentiable function $\mathrm{f}(\mathrm{x})={\int }_0^{\mathrm{x}}{\mathrm{e}}^{\mathrm{x}-\mathrm{t}}{\mathrm{f}}^{\prime }(\mathrm{t})\mathrm{dt}-\left({\mathrm{x}}^2-\mathrm{x}+1\right){\mathrm{e}}^{\mathrm{x}},\mathrm{x}\in \mathrm{R}$, is :

Let $y=y(x)$ be the solution curve of the differential equation $\frac{dy}{dx}+\frac{1}{x^2-1}y={\left(\frac{x-1}{x+1}\right)}^{\frac{1}{2}}$, $x>1$ passing through the point $\left(2,\sqrt{\frac{1}{3}}\right)$. Then $\sqrt{7}y(8)$ is equal to

The differential equation of the family of circles passing through the points $(0,2)$ and $(0,-2)$ is

Let $\mathrm{y}={\mathrm{y}}_1(\mathrm{x})$ and $\mathrm{y}={\mathrm{y}}_2(\mathrm{x})$ be two distinct solutions of the differential equation $\frac{dy}{dx}=x+y$, with ${\mathrm{y}}_1(0)=0$ and ${\mathrm{y}}_2(0)=1$ respectively. Then, the number of points of interse …

Let $y=y(x)$ be the solution curve of the differential equation
$\sin \left(2x^2\right){\log }_e\left(\tan x^2\right)dy+\left(4xy-4\sqrt{2}x\sin \left(x^2-\frac{\pi }{4}\right)\right)dx=0$,
$0<x<\sqrt{\frac{\pi }{2}}$, which passes through the point …

The slope of the tangent to a curve $C:y=y(x)$ at any point $[x,y)$ on it is $\frac{2e^{2x}-6e^{-x}+9}{2+9e^{-2x}}$. If $C$ passes through the points $\left(0,\frac{1}{2}+\frac{\pi }{2\sqrt{2}}\right)$ and …

The general solution of the differential equation $\left(x-y^2\right)dx+y\left(5x+y^2\right)dy=0$ is :

Let the solution curve $y=f(x)$ of the differential equation $\frac{dy}{dx}+\frac{xy}{x^2-1}=\frac{x^4+2x}{\sqrt{1-x^2}},x\in (-1,1)$ pass through the origin. Then ${\int }_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}}f(x)dx$ is equal to

Suppose $y=y(x)$ be the solution curve to the differential equation $\frac{dy}{dx}-y=2-e^{-x}$ such that ${\lim }_{x\to \infty }y(x)$ is finite. If $a$ and $b$ are respectively the $x$ and y-intercepts of the tangent to the curve at $\mathrm{x}=0$, then the value of …

Let $f$ be a differentiable function satisfying $f(x)=\frac{2}{\sqrt{3}}{\int }_0^{\sqrt{3}}f\left(\frac{{\lambda }^{2}x}{3}\right)d\lambda$, $x>0$ and $f(1)=\sqrt{3}$. If $y=f(x)$ passes through the point $(\alpha ,6)$, then $\alpha$ is equal to ____.

Let a smooth curve $y=f(x)$ be such that the slope of the tangent at any point ( $\mathrm{x},\mathrm{y}$ ) on it is directly proportional to $\left(\frac{-y}{x}\right)$. If the curve passes through the point $(1,2)$ and $(8,1)$, then …

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=\frac{4y^3+2y{x}^2}{3x{y}^2+x^3},y(1)=1$. If for some $n\in N,y(2)\in [n-1,n)$, then $n$ is equal to

Let f be a twice differentiable function on R. If ${\mathrm{f}}^{\prime }(0)=4$ and
$f(x)+{\int }_0^x(x-t)f^{\prime }(t)dt=\left(e^{2x}+e^{-2x}\right)\cos 2x+\frac{2}{a}x$, then $(2\mathrm{a}+1{)}^5{\mathrm{a}}^2$ is equal to

If $\frac{dy}{dx}+2y\tan x=\sin x,0<x<\frac{\pi }{2}$ and $y\left(\frac{\pi }{3}\right)=0$, then the maximum value of $\mathrm{y}(\mathrm{x})$ is

Let a curve $y=y(x)$ pass through the point $(3,3)$ and the area of the region under this curve, above the x-axis and between the abscissae 3 and $x(>3)$ be ${\left(\frac{y}{x}\right)}^3$. If this curve also passes through the point $(\alpha ,6\sqrt{10})$ in the first quadrant, t …

Let $\mathrm{x}=\mathrm{x}(\mathrm{y})$ be the solution of the differential equation $2{\mathrm{ye}}^{\mathrm{x}/{\mathrm{y}}^2}\mathrm{dx}+\left({\mathrm{y}}^2-4{\mathrm{xe}}^{\mathrm{x}/{\mathrm{y}}^2}\right)\mathrm{dy}=0$ such that $x(1)=0$. Then, $x(e)$ is equal t …

Let the slope of the tangent to a curve $y=f(x)$ at $(\mathrm{x},\mathrm{y})$ be given by $2\tan \mathrm{x}(\cos \mathrm{x}-\mathrm{y})$. if the curve passes through the point $(\pi /4,0)$, then the value of ${\int }_0^{\pi /2}ydx$ is equal to

If the solution curve of the differential equation $\left(\left({\tan }^{-1}y\right)-x\right)dy=\left(1+y^2\right)dx$ passes through the point $(1,0)$ then the abscissa of the point on the curve whose ordinate is $\tan (1)$ is :

Let v be the solution of the differential equation
$\left(1-x^2\right)dy=\left(xy+\left(x^3+2\right)\sqrt{1-x^2}\right)dx,-1<x<1$
and $y(0)=0$ if ${\int }_{-\frac{1}{2}}^{\frac{1}{2}}\sqrt{1-x^2}y(x)dx=k$ then $k^{-1}$ is equal to :

Let the solution curve $y=y(x)$ of the differential equation,

pass through the points $(1,0)$ and $(2\alpha ,\alpha ),\alpha >0$.
Then …

Let $y=y(x)$ be the solution of the differential equation $x\left(1-x^2\right)\frac{dy}{dx}+\left(3x^{2}y-y-4x^3\right)=0,x>1$, with $y(2)=-2$. Then $y(3)$ is equal to

If $y=y(x)$ is the solution of the differential equation $(1+e^{2x})\frac{dy}{dx}+2(1+y^2)e^x=0$ and $y(0)=0$, then $6\left(y^{\prime }(0)+(y({\log }_e\sqrt{3}){)}^2\right)$ is equal to:

Let $y=y(x),x>1$, be the solution of the differential equation $(x-1)\frac{dy}{dx}+2xy=\frac{1}{x-1}$, with $y(2)=\frac{1+e^4}{2e^4}$. If $y(3)=\frac{e^{\alpha }+1}{\beta e^{\alpha }}$, then the value of $\alpha +\beta$ is equal to ______.

If $y=y(x)$ is the solution of the differential equation $x\frac{dy}{dx}+2y=x{e}^x,y(1)=0$, then the local maximum value of the function $\mathrm{z}(\mathrm{x})={\mathrm{x}}^2\mathrm{y}(\mathrm{x})-{\mathrm{e}}^{\mathrm{x}}$, $\mathrm{x}\in \mathrm{R}$ is :

If the solution of the differential equation
$\frac{dy}{dx}+e^x\left(x^2-2\right)y=\left(x^2-2x\right)\left(x^2-2\right)e^{2x}$ satisfies $y(0)=0$, then the value of $y(2)$ is ____ .

Let the solution curve of the differential equation
$x\frac{dy}{dx}-y=\sqrt{y^2+16x^2},y(1)=3$ be $y=y(x)$.
Then $\mathrm{y}(2)$ is equal to :

Let $y=y(x)$ be the solution of the differential equation
$\frac{dy}{dx}+\frac{\sqrt{2}y}{2{\cos }^{4}x-\cos 2x}={\mathrm{xe}}^{{\tan }^{-1}(\sqrt{2}\cot 2\mathrm{x})},0<x<\pi /2$ with $y\left(\frac{\pi }{4}\right)=\frac{{\pi }^2}{32}$.
If …

Let the solution curve $y=y(x)$ of the differential equation $(4+x^2)dy-2x(x^2+3y+4)dx=0$ pass through the origin. Then $y(2)$ is equal to ____.

Let $S=(0,2\pi )-\left\{\frac{\pi }{2},\frac{3\pi }{4},\frac{3\pi }{2},\frac{7\pi }{4}\right\}$. Let $y=y(x)$, $x\in S$, be the solution curve of the differential equation $\frac{dy}{dx}=\frac{1}{1+\sin 2x}$, $y\left(\frac{\pi }{4}\right)=\frac{1}{2}$. If the sum of absc …

Let $\frac{dy}{dx}=\frac{ax-by+a}{bx+cy+a}$, where $a,b,c$ are constants, represent a circle passing through the point $(2,5)$. Then the shortest distance of the point $(11,6)$ from this circle is :

If ${\cos }^{-1}\left(\frac{y}{2}\right)={\log }_e{\left(\frac{x}{5}\right)}^5$, $|y|<2$, then :

If $\frac{dy}{dx}+\frac{2^{x-y}\left(2^y-1\right)}{2^x-1}=0$, $x,y>0$, $y(1)=1$, then $y(2)$ is equal to :

Let the solution curve $\mathrm{y}=\mathrm{y}(\mathrm{x})$ of the differential equation $\left(1+e^{2x}\right)\left(\frac{dy}{dx}+y\right)=1$ pass through the point $\left(0,\frac{\pi }{2}\right)$. Then, ${\lim }_{x\to \infty }{e}^{x}y(x)$ is equal to :

If $\mathrm{x}=\mathrm{x}(\mathrm{y})$ is the solution of the differential equation $y\frac{dx}{dy}=2x+y^3(y+1)e^y,x(1)=0$; then $x(e)$ is equal to :

The slope of normal at any point $(x,y),x>0,y>0$ on the curve $y=y(x)$ is given by $\frac{x^2}{xy-x^2{y}^2-1}$.
If the curve passes through the point $(1,1)$, then $e\cdot y(e)$ is equal to

If $y=y(x)$ is the solution of the differential equation $2x^2\frac{dy}{dx}-2xy+3y^2=0$ such that $y(e)=\frac{e}{3}$, then $y(1)$ is equal to

Let $y=y(x)$ be the solution of the differential equation $(x+1)y^{\prime }-y=e^{3x}(x+1{)}^2$, with $y(0)=\frac{1}{3}$. Then, the point $x=-\frac{4}{3}$ for the curve $y=y(x)$ is:

If the solution curve $y=y(x)$ of the differential equation $y^{2}dx+\left(x^2-xy+y^2\right)dy=0$, which passes through the point $(1,1)$ and intersects the line $\mathrm{y}=\sqrt{3}\mathrm{x}$ at the point $(\alpha ,\sqrt{3}\alpha )$, then value of …