Differential Equations

Let $y=y(x)$ be the solution of the differential equation $x \sqrt{1-x^2} d y+\left(y \sqrt{1-x^2}-x \cos ^{-1} x\right) d x=0, x \in(0,1), \lim _{x \rightarrow 1^{-}} y(x)=1$. Then $y\left(\frac{1}{2}\right)$ equals :

Let $y=y(x)$ be the solution of the differential equation $\left(x^2-x \sqrt{x^2-1}\right) d y+\left(y\left(x-\sqrt{x^2-1}\right)-x\right) d x=0, x \geq 1$. If $y(1)=1$, then the greatest integer less than $y(\sqrt{5})$ is $\_\_\_\_$ .

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be such that $f(x y)=f(x) f(y)$, for all $x, y \in \mathbf{R}$ and $f(0) \neq 0$. Let $g:[1, \infty) \rightarrow \mathbf{R}$ be a differentiable function such that
$$ x^2 g(x)=\int_1^x\left(\mathrm...

Let $y=y(x)$ be the solution of the differential equation $x \sin \left(\frac{y}{x}\right) d y=\left(y \sin \left(\frac{y}{x}\right)-x\right) d x, y(1)=\frac{\pi}{2}$ and let $\alpha=\cos \left(\frac{y\left(e^{12}\right)}{e^{12}}\right)$. T...

Let $f:[1, \infty) \rightarrow \mathbf{R}$ be a differentiable function defined as $f(x)=\int_1^x f(\mathrm{t}) \mathrm{dt}+(1-x)\left(\log _{\mathrm{e}} x-1\right)+\mathrm{e}$.
Then the value of $f(f(1))$ is :

Let $y=y(x)$ be the solution of the differential equation $(\tan x)^{1 / 2} \mathrm{~d} y=\left(\sec ^3 x-(\tan x)^{3 / 2} y\right) \mathrm{d} x, 0 < x <\frac{\pi}{2}, y\left(\frac{\pi}{4}\right)=\frac{6 \sqrt{2}}{5}$. If $y\left(\frac{\pi}...

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

Let $f$ be a twice differentiable function such that
\(f(x)=\int_0^x \tan (t-x) d t-\int_0^x f(t) \tan t d t, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)
Then $f^{\prime \prime}\left(\frac{\pi}{6}\right)+12 f^{\prime}\left(-\frac{\p...

Let $y=y(x)$ be the solution of the differential equation :
\(\frac{d y}{d x}+\left(\frac{6 x^2+\left(3 x^2+2 x^3+4\right) e^{-2 x}}{\left(x^3+2\right)\left(2+e^{-2 x}\right)}\right) y=2+e^{-2 x}\)
$x \in(-1,2)$, satisfying $y(0)=\frac{3}...

Let $y = y(x)$ be the solution curve of the differential equation $(1 + \sin x)\dfrac{dy}{dx} + (y + 1)\cos x = 0,\ y(0) = 0.$ If the curve $y = y(x)$ passes through the point $\left( \alpha , \dfrac{-1}{2} \right)$, then a value of $\alph...

If the curve $y = f(x)$ passes through the point $(1, e)$ and satisfies the differential equation $dy = y(2 + \\log_e x) dx$, $x > 0$, then $f(e)$ is equal to :

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

Let $y=y(x)$ be the solution of the differential equation
\(x \frac{d y}{d x}-\sin 2 y=x^3\left(2-x^3\right) \cos ^2 y, x \neq 0 .\)
If $y(2)=0$, then $\tan (y(1))$ is equal to

Let $y = y(x)$ be the solution of the differential equation $x \frac{dy}{dx} - y = x^2 \cot x$, $x \in (0, \pi)$. If $y\left(\frac{\pi}{2}\right) = \frac{\pi}{2}$, then$6y\left(\frac{\pi}{6}\right) - 8y\left(\frac{\pi}{4}\right)$ is equal t...

Let $f$ be a twice differentiable non-negative function such that $(f(x))^2=25+\int_0^x\left((f(\mathrm{t}))^2+\left(f^{\prime}(\mathrm{t})\right)^2\right) \mathrm{dt}$. Then the mean of $f\left(\log _{\mathrm{e}}(1)\right), f\left(\log _{\...

Let $y=y(x)$ be the solution of the differential equation $x^4 \mathrm{~d} y+\left(4 x^3 y+2 \sin x\right) \mathrm{d} x=0, x>0, y\left(\frac{\pi}{2}\right)=0$.
Then $\pi^4 y\left(\frac{\pi}{3}\right)$ is equal to :

If the solution curve $y=f(x)$ of the differential equation
\(\left(x^2-4\right) y^{\prime}-2 x y+2 x\left(4-x^2\right)^2=0, x>2,\)
passes through the point $(3,15)$, then the local maximum value of $f$ is $\_\_\_\_$

Let the solution curve of the differential equation $x d y-y d x=\sqrt{x^2+y^2} d x, x>0$, $y(1)=0$, be $y=y(x)$. Then $y(3)$ is equal to

If $y=y(x)$ satisfies the differential equation $16(\sqrt{x+9 \sqrt{x}})(4+\sqrt{9+\sqrt{x}}) \cos y \mathrm{~d} y=(1+2 \sin y) \mathrm{d} x, x>0$ and $y(256)=\frac{\pi}{2}, y(49)=\alpha$, then $2 \sin \alpha$ is equal to :

Let $y=y(x)$ be the solution curve of the differential equation $\left(1+x^2\right) \mathrm{d} y+\left(y-\tan ^{-1} x\right) d x=0, y(0)=1$. Then the value of $y(1)$ is :

Let $y = y(x)$ be the solution of the differential equation $\sec x \dfrac{dy}{dx} - 2y = 2 + 3 \sin x$, $x \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$,$y(0) = -\dfrac{7}{4}$. Then $y\left(\dfrac{\pi}{6}\right)$ is equal to :

Let $f(x) = x - 1$ and $g(x) = e^x$ for $x \in \mathbb{R}$. If $\frac{dy}{dx} = \left( e^{-2\sqrt{x}} g\left(f(f(x))\right) - \frac{y}{\sqrt{x}} \right)$, $y(0) = 0$, then $y(1)$ is

Let $y=y(x)$ be the solution curve of the differential equation
$x\left(x^2+e^x\right) d y+\left(\mathrm{e}^x(x-2) y-x^3\right) \mathrm{d} x=0, x>0$, passing through the point $(1,0)$. Then $y(2)$ is equal to :

Let y = y(x) be the solution of the differential equation $(x^2 + 1)y' - 2xy = (x^4 + 2x^2 + 1)\cos x$,
$y(0) = 1$. Then $ \int\limits_{-3}^{3} y(x) \, dx $ is :

If a curve $y=y(x)$ passes through the point $\left(1, \frac{\pi}{2}\right)$ and satisfies the differential equation $\left(7 x^4 \cot y-\mathrm{e}^x \operatorname{cosec} y\right) \frac{\mathrm{d} x}{\mathrm{~d} y}=x^5, x \geq 1$, then at $...

Let $g$ be a differentiable function such that $\int_0^x g(t) d t=x-\int_0^x \operatorname{tg}(t) d t, x \geq 0$ and let $y=y(x)$ satisfy the differential equation $\frac{d y}{d x}-y \tan x=2(x+1) \sec x g(x), x \in\left[0, \frac{\pi}{2}\ri...

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}+3\left(\tan ^2 x\right) y+3 y=\sec ^2 x, y(0)=\frac{1}{3}+e^3$. Then $y\left(\frac{\pi}{4}\right)$ is equal to :

Let $y=y(x)$ be the solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+2 y \sec ^2 x=2 \sec ^2 x+3 \tan x \cdot \sec ^2 x$ such that $y(0)=\frac{5}{4}$. Then $12\left(y\left(\frac{\pi}{4}\right)-\mathrm{e}^{-2}\right)...

Let y = y(x) be the solution of the differential equation :
$\cos x\left(\log _e(\cos x)\right)^2 d y+\left(\sin x-3 y \sin x \log _e(\cos x)\right) d x=0$, x ∈ (0, $\frac{\pi}{2}$ ). If $ y(\frac{\pi}{4}) $ = $-\frac{1}{\log_{e}2}$, then $...

If for the solution curve $y=f(x)$ of the differential equation $\frac{d y}{d x}+(\tan x) y=\frac{2+\sec x}{(1+2 \sec x)^2}$, $x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right), f\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{10}$, then $f\left(\...

Let for some function $\mathrm{y}=f(x), \int_0^x t f(t) d t=x^2 f(x), x>0$ and $f(2)=3$. Then $f(6)$ is equal to

If $y=y(x)$ is the solution of the differential equation,
$\sqrt{4-x^2} \frac{\mathrm{~d} y}{\mathrm{~d} x}=\left(\left(\sin ^{-1}\left(\frac{x}{2}\right)\right)^2-y\right) \sin ^{-1}\left(\frac{x}{2}\right),-2 \leq x \leq 2, y(2)=\frac{\pi...

Let $f$ be a differentiable function such that $2(x+2)^2 f(x)-3(x+2)^2=10 \int_0^x(t+2) f(t) d t, x \geq 0$. Then $f(2)$ is equal to ________ .

Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $\left(x y-5 x^2 \sqrt{1+x^2}\right) d x+\left(1+x^2\right) d y=0, y(0)=0$. Then $y(\sqrt{3})$ is equal to

Let $y=y(x)$ be the solution of the differential equation $2 \cos x \frac{\mathrm{~d} y}{\mathrm{~d} x}=\sin 2 x-4 y \sin x, x \in\left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{3}\right)=0$, then $y^{\prime}\left(\frac{\pi}{4}\right)...

Let a curve $y=f(x)$ pass through the points $(0,5)$ and $\left(\log _e 2, k\right)$. If the curve satisfies the differential equation $2(3+y) e^{2 x} d x-\left(7+e^{2 x}\right) d y=0$, then $k$ is equal to

Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y \frac{\mathrm{~d} x}{\mathrm{~d} y}\right) \sin \left(\frac{x}{y}\right), y>0$ and $x(1)=\frac{\pi}{2}$. Then $\cos (x(2))$ is equal to :

Let $f(x)$ be a real differentiable function such that $f(0)=1$ and $f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(y)$ for all $x, y \in \mathbf{R}$. Then $\sum_\limits{n=1}^{100} \log _e f(n)$ is equal to :

Let $x=x(y)$ be the solution of the differential equation $y^2 \mathrm{~d} x+\left(x-\frac{1}{y}\right) \mathrm{d} y=0$. If $x(1)=1$, then $x\left(\frac{1}{2}\right)$ is :

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $f(x+y)=f(x) f(y)$ for all $x, y \in \mathbf{R}$. If $f^{\prime}(0)=4 \mathrm{a}$ and $f$ satisfies $f^{\prime \prime}(x)-3 \mathrm{a} f^{\prime}(x)-f(x...

Let $y=f(x)$ be the solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{x y}{x^2-1}=\frac{x^6+4 x}{\sqrt{1-x^2}},-1< x<1$ such that $f(0)=0$. If $6 \int_{-1 / 2}^{1 / 2} f(x) \mathrm{d} x=2 \pi-\alpha$ then $\alp...

If $x=f(y)$ is the solution of the differential equation $\left(1+y^2\right)+\left(x-2 \mathrm{e}^{\tan ^{-1} y}\right) \frac{\mathrm{d} y}{\mathrm{~d} x}=0, y \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ with $f(0)=1$, then $f\left(\frac...

The solution curve, of the differential equation \(2 y \frac{\mathrm{d} y}{\mathrm{~d} x}+3=5 \frac{\mathrm{d} y}{\mathrm{~d} x}\), passing through the point \((0,1)\) is a conic, whose vertex lies on the line :

The solution of the differential equation \((x^2+y^2) \mathrm{d} x-5 x y \mathrm{~d} y=0, y(1)=0\), is :

For a differentiable function \(f: \mathbb{R} \rightarrow \mathbb{R}\), suppose \(f^{\prime}(x)=3 f(x)+\alpha\), where \(\alpha \in \mathbb{R}, f(0)=1\) and \(\lim _\limits{x \rightarrow-\infty} f(x)=7\). Then \(9 f\left(-\log _e 3\right)\)...

Let \(\int_\limits0^x \sqrt{1-\left(y^{\prime}(t)\right)^2} d t=\int_0^x y(t) d t, 0 \leq x \leq 3, y \geq 0, y(0)=0\). Then at \(x=2, y^{\prime \prime}+y+1\) is equal to

Let \(y=y(x)\) be the solution of the differential equation \((1+y^2) e^{\tan x} d x+\cos ^2 x(1+e^{2 \tan x}) d y=0, y(0)=1\). Then \(y\left(\frac{\pi}{4}\right)\) is equal to

Let \(f(x)\) be a positive function such that the area bounded by \(y=f(x), y=0\) from \(x=0\) to \(x=a>0\) is \(e^{-a}+4 a^2+a-1\). Then the differential equation, whose general solution is \(y=c_1 f(x)+c_2\), where \(c_1\) and \(c_2\) are...

Let \(\alpha|x|=|y| \mathrm{e}^{x y-\beta}, \alpha, \beta \in \mathbf{N}\) be the solution of the differential equation \(x \mathrm{~d} y-y \mathrm{~d} x+x y(x \mathrm{~d} y+y \mathrm{~d} x)=0,y(1)=2\). Then \(\alpha+\beta\) is equal to ___...

Let \(y=y(x)\) be the solution curve of the differential equation \(\sec y \frac{\mathrm{d} y}{\mathrm{~d} x}+2 x \sin y=x^3 \cos y, y(1)=0\). Then \(y(\sqrt{3})\) is equal to: