Differential Equations — JEE Advanced PYQs

Let $\mathrm{y}:(-\infty ,\infty )\to (0,\infty )$ be the solution of the differential equation

satisfying $\mathrm{y}(0)=\frac{1}{\sqrt{2}}$. Then the value of $\mathrm{y}\left({\log }_{\mathrm{e}}2\right)$ …

Let $\mathrm{y}=f(\mathrm{x})$ be the real valued function defined on the interval $(0,\infty )$, satisfying $\mathrm{y}(1)=0$ and the differential equation

Then which of the following statements is (are) TRUE ?
(A) The function $f$ has a local minim …

For all $x>0$, let $y_1(x),y_2(x)$, and $y_3(x)$ be the functions satisfying …

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

satisfying $y(1)=0$. Then the value of $2\frac{(y(e){)}^2}{y\left(e^2\right)}$ is ____.

Let $f(\mathrm{x})$ be a continuously differentiable function on the interval $(0,\infty )$ such that $f(1)=2$ and ${\lim }_{\mathrm{t}\to \mathrm{x}}\frac{{\mathrm{t}}^{10}f(\mathrm{x})-{\mathrm{x}}^{10}f(\mathrm{t})}{{\mathrm{t}}^9-{\mathrm{x}}^9}=1$ for each …

Let $\mathrm{f}:[1,\infty )\to \mathbb{R}$ be a differentiable function such that $\mathrm{f}(1)=\frac{1}{3}$ and $3{\int }_1^{\mathrm{x}}\mathrm{f}(\mathrm{t})\mathrm{dt}=\mathrm{xf}(\mathrm{x})-\frac{{\mathrm{x}}^3}{3},\mathrm{x}\in [1,\infty )$. Let $e$ denote the …

For $\mathrm{x}\in \mathbb{R}$, let $\mathrm{y}(\mathrm{x})$ be a solution of the differential equation $\left({\mathrm{x}}^2-5\right)\frac{\mathrm{dy}}{\mathrm{dx}}-2\mathrm{xy}=-2\mathrm{x}{\left({\mathrm{x}}^2-5\right)}^2$ such that $\mathrm{y}(2)=7$.
Then the maximum valu …

If $y(x)$ is the solution of the differential equation

and the slope of the curve $y=y(x)$ is never zero, then the value of $10y(\sqrt{2})$ is ____.

For $x\in \mathbb{R}$, let the function $y(x)$ be the solution of the differential equation

Then, which of the following statements is/are TRUE?

For any real numbers $\alpha$ and $\beta$, let ${\mathrm{y}}_{\alpha ,\beta }(\mathrm{x}),\mathrm{x}\in \mathbb{R}$, be the solution of the differential equation

Let …

Let $b$ be a nonzero real number. Suppose $f:\mathbb{R}\to \mathbb{R}$ is a differentiable function such that $f(0)=1$.
If the derivative $f^{\prime }$ of $f$ satisfies the equation $f^{\prime }(x)=\frac{f(x)}{b^2+x^2}$
for all $x\in \mathbb{R}$, then w …

Let $\Gamma$ denote a curve $\mathrm{y}=\mathrm{y}(\mathrm{x})$ which is in the first quadrant and let the point $(1,0)$ lie on it. Let the tangent to $\Gamma$ at a point P intersect the y-axis at ${\mathrm{Y}}_{\mathrm{P}}$. If ${\mathrm{PY}}_{\mathrm{P}}$ has length 1 for each poin …

Let $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ be two non-constant differentiable functions. If $f^{\prime }(x)=\left(e^{(f(x)-g(x))}\right)g^{\prime }(x)$ for all $x\in \mathbb{R}$, and $f(1)=g(2)=1$, then which of the following s …

Let $f:[0,\infty )\to \mathbb{R}$ be a continuous function such that $f(x)=1-2x+{\int }_0^x{e}^{x-t}f(t)dt$ for all $x\in [0,\infty )$. Then, which of the following statement(s) is (are) TRUE?

Let $f:(0,\pi )\to \mathbb{R}$ be a twice differentiable function such that

If $f\left(\frac{\pi }{6}\right)=-\frac{\pi }{12}$, then which of the following s …

Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function with $f(0)=0$. If $y=f(x)$ satisfies the differential equation

then the value of ${\lim }_{x\to -\infty }f(x)$ is ____.

Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function with $f(0)=1$ and satisfying the equation

Then, the value of ${\log }_e(f(4))$ is ____.

If $y=y(x)$ satisfies the differential equation

and $y(0)=\sqrt{7}$, then $y(256)=$

Let $f:(0,\infty )\to \mathbb{R}$ be a differentiable function such that $f^{\prime }(x)=2-\frac{f(x)}{x}$ for all $x\in (0,\infty )$ and $f(1)\ne 1$. Then

A solution curve of the differential equation $\left(x^2+xy+4x+2y+4\right)\frac{dy}{dx}-y^2=0,x>0$, passes through the point (1, 3). Then the solution curve

Let $y(x)$ be a solution of the differential equation $\left(1+e^x\right)y^{\prime }+y{e}^x=1$. If $y(0)=2$, then which of the following statements is (are) true?

Consider the family of all circles whose centers lie on the straight line $y=x$. If this family of circles is represented by the differential equation $P{y}^{\prime \prime }+Q{y}^{\prime }+1=0$, where $P,Q$ are functions of $x,y$ and $y^{\prime }$ (here …

The function $y=f(x)$ is the solution of the differential equation

in $(-1,1)$ satisfying $f(0)=0$. Then

is

A curve passes through the point $\left(1,\frac{\pi }{6}\right)$. Let the slope of the curve at each point $(x,y)$ be $\frac{y}{x}+\sec \left(\frac{y}{x}\right),x>0$. Then the equation of the curve is

Let $f:[0,1]\to \mathbb{R}$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable, $f(0)=f(1)=0$ and satisfies $f^{\prime \prime }(x)-2f^{\prime }(x)+f(x)\ge e^x,x\in [0,1]$.

Which of the following is true for $0<x<1$ ?

If $y(x)$ satisfies the differential equation $y^{\prime }-y\tan x=2x\sec x$ and $y(0)=0$, then

Let $y^{\prime }(x)+y(x)g^{\prime }(x)=g(x)g^{\prime }(x),y(0)=0,x\in \mathbb{R}$, where $f^{\prime }(x)$ denotes $\frac{df(x)}{dx}$ and $g(x)$ is a given non-constant differentiable function on $\mathbb{R}$ with $g(0)=g(2)=0$. Then the value of …

Let f be a real-valued differentiable function on R (the set of all real numbers) such that $\mathrm{f}(1)=1$. If the y-intercept of the tangent at any point $\mathrm{P}(\mathrm{x},\mathrm{y})$ on the curve $\mathrm{y}=\mathrm{f}(\mathrm{x})$ is equal to the cube of the ab …

Let $f:\mathbf{R}\to \mathbf{R}$ be a continuous function which satisfies

Then the value of $f(\ln 5)$ is

Let $f$ be a non-negative function defined on the interval $[0,1]$. If

and $f(0)=0$, then

Let $f(x)$ be differentiable on the interval $(0,\infty )$ such that $f(1)=1$, and

for each $x>0$. Then $f(x)$ is

The differential equation $\frac{dy}{dx}=\frac{\sqrt{1-y^2}}{y}$ determines a family of circles with

If $f^{\prime \prime }(x)=-f(x)$ and $g(x)=f^{\prime }(x)$ and $F(x)={\left(f\left(\frac{x}{2}\right)\right)}^2+{\left(g\left(\frac{x}{2}\right)\right)}^2$ and given that $F(5)=5$, then $F(10)$ is equal to

A tangent drawn to the curve $y=f(x)$ at $P(x,y)$ cuts the x-axis and y-axis at A and B respectively such that BP : AP $=3:1$, given that $f(1)=1$, then