Differential Equations — JEE Main PYQs

Let $y=y(x)$ be a solution curve of the differential equation $(y+1){\tan }^{2}xdx+\tan xdy+ydx=0,x\in \left(0,\frac{\pi }{2}\right)$. If ${\lim }_{x\to 0^{+}}xy(x)=1$, then the value of $y\left(\frac{\pi }{4}\right)$ is:

The population $\mathrm{P}=\mathrm{P}(\mathrm{t})$ at time 't' of a certain species follows the differential equation $\frac{\mathrm{dP}}{\mathrm{dt}}=0.5\mathrm{P}-450$. If $P(0)=850$, then the time at which population becomes zero is:

Let $y(x)$ be the solution of the differential equation $2x^{2}dy+\left(e^y-2x\right)dx=0,x>0$. If $y(e)=1$, then $y(1)$ is equal to:

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

Let $y=y(x)$ be the solution of the differential equation
$x\tan \left(\frac{y}{x}\right)dy=\left(y\tan \left(\frac{y}{x}\right)-x\right)dx,-1\le x\le 1,y\left(\frac{1}{2}\right)=\frac{\pi }{6}$
Then the area of the region bounded by the curves …

Let $f$ be a twice differentiable function defined on $R$ such that $f(0)=1,f^{\prime }(0)=2$ and $f(x)\ne 0$ for all $x\in R$. If $\left|\begin{array}{ll}f(x)& f^{\prime }(x)\\ f^{\prime }(x)& f^{\prime \prime }(x)\end{array}\right|=0$, for all $x\in R$, then the value of …

If a curve $y=f(x)$ passes through the point $(1,2)$ and satisfies $x\frac{dy}{dx}+y=b{x}^4$, then for what value of $b,{\int }_1^{2}f(x)dx=\frac{62}{5}$ ?

Let $y=y(x)$ satisfies the equation $\frac{dy}{dx}-|A|=0$, for all $x>0$, where $A=\left[\begin{array}{ccc}y& \sin x& 1\\ 0& -1& 1\\ 2& 0& \frac{1}{x}\end{array}\right]$. If $y(\pi )=\pi +2$, then the value of $y\left(\frac{\pi }{2}\right)$ is :

Let a curve $y=y(x)$ be given by the solution of the differential equation $\cos \left(\frac{1}{2}{\cos }^{-1}\left(e^{-x}\right)\right)dx=\sqrt{e^{2x}-1}dy$
If it intersects $y$-axis at $y=-1$, and the intersection point of the curve with $x$-axis is $(\alpha ,0)$, then …

A differential equation representing the family of parabolas with axis parallel to $y$-axis and whose length of latus rectum is the distance of the point $(2,-3)$ form the line $3x+4y=5$, is given by :

If the solution curve of the differential equation $\left(2x-10y^3\right)dy+ydx=0$, passes through the points $(0,1)$ and $(2,\beta )$, then $\beta$ is a root of the equation:

If a curve passes through the origin and the slope of the tangent to it at any point $(x,y)$ is $\frac{x^2-4x+y+8}{x-2}$, then this curve also passes through the point :

Let $y=y(x)$ be the solution of the differential equation ${\mathrm{cosec}}^{2}xdy+2dx=(1+y\cos 2x){\mathrm{cosec}}^{2}xdx$, with $y\left(\frac{\pi }{4}\right)=0$. Then, the value of $(y(0)+1{)}^2$ is equal to :

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

$\left((x+2)e^{\left(\frac{y+1}{x+2}\right)}+(y+1)\right)dx=(x+2)dy$, $y(1)=1$.

If the domain of $y=y(x)$ is an open interval $(\alpha ,\beta )$, then $|\alpha +\beta |$ is equal to ____

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time $t=0$. The number of bacteria is increased by $20\%$ in 2 hours. If the population of bacteria is 2000 after …

If $y=y(x)$ is the solution of the equation $e^{\sin y}\cos y\frac{dy}{dx}+e^{\sin y}\cos x=\cos x,y(0)=0$; then $1+y\left(\frac{\pi }{6}\right)+\frac{\sqrt{3}}{2}y\left(\frac{\pi }{3}\right)+\frac{1}{\sqrt{2}}y\left(\frac{\pi }{4}\right)$ is equal to ____

The difference between degree and order of a differential equation that represents the family of curves given by $y^2=a\left(x+\frac{\sqrt{a}}{2}\right),a>0$ is ______

Let us consider a curve, $y=f(x)$ passing through the point $(-2,2)$ and the slope of the tangent to the curve at any point $(x,f(x))$ is given by $f(x)+x{f}^{\prime }(x)=x^2$. Then:

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=2(y+2\sin x-5)x-2\cos x$ such that that $y(0)=7$. Then $y(\pi )$ is equal to

If $y^{1/4}+y^{-1/4}=2x$ and $\left(x^2-1\right)\frac{d^{2}y}{d{x}^2}+\alpha x\frac{dy}{dx}+\beta y=0$, then $|\alpha -\beta |$ is equal to

Let $f(x)={\int }_0^x{e}^{t}f(t)dt+e^x$ be a differentiable function for all $x\in R$. Then $f(x)$ equals :

Let slope of the tangent line to a curve at any point $P(x,y)$ be given by $\frac{x{y}^2+y}{x}$. If the curve intersects the line $x+2y=4$ at $x=-2$, then the value of $y$, for which the point ( $3,y$ ) lies on the curve, is :

If $y\frac{dy}{dx}=x\left[\frac{y^2}{x^2}+\frac{\varphi \left(\frac{y^2}{x^2}\right)}{{\varphi }^{\prime }\left(\frac{y^2}{x^2}\right)}\right],x>0,\varphi >0$, and $y(1)=-1$, then $\varphi \left(\frac{y^2}{4}\right)$ is equal to :

Let the normals at all the points on a given curve pass through a fixed point (a, b). If the curve passes through $(3,-3)$ and $(4,-2\sqrt{2})$ and given that $a-2\sqrt{2}b=3$, then $\left(a^2+b^2+ab\right)$ is equal to ____

If $\frac{dy}{dx}=\frac{2^{x}y+2^y\cdot 2^x}{2^x+2^{x+y}{\log }_{e}2},y(0)=0$, then for $y=1$, the value of $x$ lies in the interval:

If , then is equal to:

Let $y=y(x)$ be solution of the differential equation ${\log }_e\left(\frac{dy}{dx}\right)=3x+4y$, with $y(0)=0$. If $y\left(-\frac{2}{3}{\log }_{e}2\right)=\alpha {\log }_{e}2$, then the value of $\alpha$ is equal to :

If $y=y(x),y\in \left[0,\frac{\pi }{2}\right]$ is the solution of the differential equation $\sec y\frac{dy}{dx}-\sin (x+y)-\sin (x-y)=0$, with $y(0)=0$, then $5y^{\prime }\left(\frac{\pi }{2}\right)$ is equal to ____ .

Which of the following is true for $y(x)$ that satisfies the differential equation $\frac{dy}{dx}=xy-1+x-y$; $y(0)=0$:

Let $y=y(x)$ be the solution of the differential equation
$\frac{dy}{dx}=1+x{e}^{y-x},-\sqrt{2}<x<\sqrt{2},y(0)=0$
then, the minimum value of $y(x),x\in (-\sqrt{2},\sqrt{2})$ is equal to :

Let $y=y(x)$ be solution of the following differential equation $e^y\frac{dy}{dx}-2e^y\sin x+\sin x{\cos }^{2}x=0,y\left(\frac{\pi }{2}\right)=0$.
If $y(0)={\log }_e\left(\alpha +\beta e^{-2}\right)$, then $4(\alpha +\beta )$ is equal to ____

Let $y=y(x)$ be the solution of the differential equation $xdy=\left(y+x^3\cos x\right)dx$ with $y(\pi )=0$, then $y\left(\frac{\pi }{2}\right)$ is equal to :

Let a curve $y=f(x)$ pass through the point $\left(2,{\left({\log }_{e}2\right)}^2\right)$ and have slope $\frac{2y}{x{\log }_{e}x}$ for all positive real value of $x$. Then the value of $f(e)$ is equal to ____

Let $y=y(x)$ be the solution of the differential equation
$(x-x^3)dy=(y+y{x}^2-3x^4)dx,x>2$. If $y(3)=3$ then $y(4)$ is equal to

Let $y=y(x)$ be the solution of the differential equation $dy=e^{\alpha x+y}dx;\alpha \in N$. If $y\left({\log }_{e}2\right)={\log }_{e}2$ and $y(0)={\log }_e\left(\frac{1}{2}\right)$, then the value of $\alpha$ is equal to ______

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

If $y=y(x)$ is the solution of the differential equation, $\frac{dy}{dx}+2y\tan x=\sin x,y\left(\frac{\pi }{3}\right)=0$, then the maximum value of the function $y(x)$ over $R$ is equal to :

Let the curve $y=y(x)$ be the solution of the differential equation, $\frac{dy}{dx}=2(x+1)$. If the numerical value of area bounded by the curve $y=y(x)$ and $x$-axis is $\frac{4\sqrt{8}}{3}$, then the value of $y(1)$ is equal to ____

If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx}+(\tan x)y=\sin x,0\le x\le \frac{\pi }{3}$, with $y(0)=0$, then $y\left(\frac{\pi }{4}\right)$ equal to:

Let ${\mathrm{C}}_1$ be the curve obtained by the solution of differential equation $2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}}={\mathrm{y}}^2-{\mathrm{x}}^2,\mathrm{x}>0$. Let the curve $C_2$ be the solution of $\frac{2xy}{x^2-y^2}=\frac{dy}{dx}$. If both t …

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=(y+1)\left((y+1)e^{x^2/2}-x\right),0<x<2.1$, with $y(2)=0$. Then the value of $\frac{dy}{dx}$ at $x=1$ is equal to:

Let $y=y(x)$ be the solution of the differential equation $xdy-ydx=\sqrt{(x^2-y^2)}dx,x\ge 1$, with $y(1)=0$. If the area bounded by the line $x=1,x=e^{\pi },y=0$ and $y=y(x)$ is $\alpha e^{2\pi }+\beta$, then the value of $10(\alpha +\beta )$ is equal to......

If the curve $y=y(x)$ is the solution of the differential equation
$2\left(x^2+x^{5/4}\right)dy-y\left(x+x^{1/4}\right)dx=2x^{9/4}dx,x>0$ which passes through the point $\left(1,1-\frac{4}{3}{\log }_{e}2\right)$, then the value of $y(16)$ is equal to:

Let $y=y(x)$ be in the solution of the differential equation $\cos x(3\sin x+\cos x+3)dy=(1+y\sin x(3\sin x+\cos x+3))dx,0\le x\le \frac{\pi }{2},y(0)=0$. Then $y\left(\frac{\pi }{3}\right)$ is equal to:

The differential equation satisfied by the system of parabolas $y^2=4a(x+a)$ is :

If $\mathrm{y}=\mathrm{y}(\mathrm{x})$ is the solution of the differential equation, ${\mathrm{e}}^{\mathrm{y}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}-1\right)={\mathrm{e}}^{\mathrm{x}}$ such that $y(0)=0$, then $y(1)$ is equal to:

Let $y=y(x)$ be the solution curve of the differential equation, $\left(y^2-x\right)\frac{dy}{dx}=1$, satisfying $y(0)=1$. This curve intersects the $x$-axis at a point whose abscissa is:

Let $y=y(x)$ be a solution of the differential equation, $\sqrt{1-x^2}\frac{dy}{dx}+\sqrt{1-y^2}=0,|x|<1$. If $\mathrm{y}\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}$, then $\mathrm{y}\left(\frac{1}{\sqrt{2}}\right)$ is equal to:

The differential equation of the family of curves, $x^2=4b(y+b),b\in R$, is:

If $\frac{dy}{dx}=\frac{xy}{x^2+y^2};y(1)=1$; then a value of $x$ satisfying $y(x)=e$ is: