Differential Equations

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

Let \(y=y(x)\) be the solution of the differential equation \(\left(2 x \log _e x\right) \frac{d y}{d x}+2 y=\frac{3}{x} \log _e x, x>0\) and \(y\left(e^{-1}\right)=0\). Then, \(y(e)\) is equal to

If the solution \(y(x)\) of the given differential equation \(\left(e^y+1\right) \cos x \mathrm{~d} x+\mathrm{e}^y \sin x \mathrm{~d} y=0\) passes through the point \(\left(\frac{\pi}{2}, 0\right)\), then the value of $$e^{y\left(\frac{\pi}...

Suppose the solution of the differential equation \(\frac{d y}{d x}=\frac{(2+\alpha) x-\beta y+2}{\beta x-2 \alpha y-(\beta \gamma-4 \alpha)}\) represents a circle passing through origin. Then the radius of this circle is :

If \(y=y(x)\) is the solution of the differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}+2 y=\sin (2 x), y(0)=\frac{3}{4}\), then \(y\left(\frac{\pi}{8}\right)\) is equal to :

Let \(y=y(x)\) be the solution of the differential equation
\(\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{2 x}{\left(1+x^2\right)^2} y=x \mathrm{e}^{\frac{1}{\left(1+x^2\right)}} ; y(0)=0.\)
Then the area enclosed by the curve $$f(x)=y(x) \mat...

The differential equation of the family of circles passing through the origin and having centre at the line \(y=x\) is :

Let the solution \(y=y(x)\) of the differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}-y=1+4 \sin x\) satisfy \(y(\pi)=1\). Then \(y\left(\frac{\pi}{2}\right)+10\) is equal to __________.

If the solution \(y=y(x)\) of the differential equation \((x^4+2 x^3+3 x^2+2 x+2) \mathrm{d} y-(2 x^2+2 x+3) \mathrm{d} x=0\) satisfies \(y(-1)=-\frac{\pi}{4}\), then \(y(0)\) is equal to :

Let \(y=y(x)\) be the solution of the differential equation \((x^2+4)^2 d y+(2 x^3 y+8 x y-2) d x=0\). If \(y(0)=0\), then \(y(2)\) is equal to

Let \(y=y(x)\) be the solution of the differential equation \((x+y+2)^2 d x=d y, y(0)=-2\). Let the maximum and minimum values of the function \(y=y(x)\) in \(\left[0, \frac{\pi}{3}\right]\) be \(\alpha\) and \(\beta\), respectively. If $$(...

The solution curve of the differential equation
\(y \frac{d x}{d y}=x\left(\log _e x-\log _e y+1\right), x>0, y>0\) passing through the point \((e, 1)\) is

Let \(y=y(x)\) be the solution of the differential equation \(\frac{d y}{d x}=\frac{(\tan x)+y}{\sin x(\sec x-\sin x \tan x)}, x \in\left(0, \frac{\pi}{2}\right)\) satisfying the condition \(y\left(\frac{\pi}{4}\right)=2\). Then, $$y\left(\...

Let \(y=y(x)\) be the solution of the differential equation
\(\sec ^2 x d x+\left(e^{2 y} \tan ^2 x+\tan x\right) d y=0,0< x<\frac{\pi}{2}, y(\pi / 4)=0\).
If \(y(\pi / 6)=\alpha\), then \(e^{8 \alpha}\) is equal to ____________.

The temperature \(T(t)\) of a body at time \(t=0\) is \(160^{\circ} \mathrm{F}\) and it decreases continuously as per the differential equation \(\frac{d T}{d t}=-K(T-80)\), where \(K\) is a positive constant. If $$T(15)=120^{\circ} \mathrm...

Let \(y=y(x)\) be the solution of the differential equation \(\left(1-x^2\right) \mathrm{d} y=\left[x y+\left(x^3+2\right) \sqrt{3\left(1-x^2\right)}\right] \mathrm{d} x, -1< x<1, y(0)=0\). If $$y\left(\frac{1}{2}\right)=\frac{\mathrm{m}}{\...

Let \(y=y(x)\) be the solution of the differential equation \(\sec x \mathrm{~d} y+\{2(1-x) \tan x+x(2-x)\} \mathrm{d} x=0\) such that \(y(0)=2\). Then \(y(2)\) is equal to:

Let \(Y=Y(X)\) be a curve lying in the first quadrant such that the area enclosed by the line \(Y-y=Y^{\prime}(x)(X-x)\) and the co-ordinate axes, where \((x, y)\) is any point on the curve, is always $$\frac{-y^2}{2 Y^{\prime}(x)}+1, Y^{\p...

If the solution curve \(y=y(x)\) of the differential equation \(\left(1+y^2\right)\left(1+\log _{\mathrm{e}} x\right) d x+x d y=0, x > 0\) passes through the point \((1,1)\) and $$y(e)=\frac{\alpha-\tan \left(\frac{3}{2}\right)}{\beta+\tan ...

A function \(y=f(x)\) satisfies \(f(x) \sin 2 x+\sin x-\left(1+\cos ^2 x\right) f^{\prime}(x)=0\) with condition \(f(0)=0\). Then, \(f\left(\frac{\pi}{2}\right)\) is equal to

If \(\sin \left(\frac{y}{x}\right)=\log _e|x|+\frac{\alpha}{2}\) is the solution of the differential equation \(x \cos \left(\frac{y}{x}\right) \frac{d y}{d x}=y \cos \left(\frac{y}{x}\right)+x\) and \(y(1)=\frac{\pi}{3}\), then $$\alpha^2$...

If the solution of the differential equation $(2 x+3 y-2) \mathrm{d} x+(4 x+6 y-7) \mathrm{d} y=0, y(0)=3$, is
$\alpha x+\beta y+3 \log _e|2 x+3 y-\gamma|=6$, then $\alpha+2 \beta+3 \gamma$ is equal to ____________.

Let $x=x(\mathrm{t})$ and $y=y(\mathrm{t})$ be solutions of the differential equations $\frac{\mathrm{d} x}{\mathrm{dt}}+\mathrm{a} x=0$ and $\frac{\mathrm{d} y}{\mathrm{dt}}+\mathrm{by}=0$ respectively, $\mathrm{a}, \mathrm{b} \in \mathbf{...

If the solution curve, of the differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{x+y-2}{x-y}\) passing through the point \((2,1)\) is $$\tan ^{-1}\left(\frac{y-1}{x-1}\right)-\frac{1}{\beta} \log _{\mathrm{e}}\left(\alpha+\lef...

If \(y=y(x)\) is the solution curve of the differential equation \(\left(x^2-4\right) \mathrm{d} y-\left(y^2-3 y\right) \mathrm{d} x=0, x>2, y(4)=\frac{3}{2}\) and the slope of the curve is never zero, then the value of \(y(10)\) equals :

If $x=x(t)$ is the solution of the differential equation $(t+1) \mathrm{d} x=\left(2 x+(t+1)^4\right) \mathrm{dt}, x(0)=2$, then, $x(1)$ equals _________.

Let $y=y(x)$ be the solution of the differential equation
$\frac{\mathrm{d} y}{\mathrm{~d} x}=2 x(x+y)^3-x(x+y)-1, y(0)=1$.
Then, $\left(\frac{1}{\sqrt{2}}+y\left(\frac{1}{\sqrt{2}}\right)\right)^2$ equals :

If $\frac{\mathrm{d} x}{\mathrm{~d} y}=\frac{1+x-y^2}{y}, x(1)=1$, then $5 x(2)$ is equal to __________.

Let $\alpha$ be a non-zero real number. Suppose $f: \mathbf{R} \rightarrow \mathbf{R}$ is a differentiable function such that $f(0)=2$ and $\lim\limits_{x \rightarrow-\infty} f(x)=1$. If $f^{\prime}(x)=\alpha f(x)+3$, for all $x \in \mathbf...

If the solution curve of the differential equation \(\left(y-2 \log _{e} x\right) d x+\left(x \log _{e} x^{2}\right) d y=0, x > 1\) passes through the points \(\left(e, \frac{4}{3}\right)\) and \(\left(e^{4}, \alpha\right)\), then $$\alpha$...

Let the solution curve \(x=x(y), 0 < y < \frac{\pi}{2}\), of the differential equation \(\left(\log _{e}(\cos y)\right)^{2} \cos y \mathrm{~d} x-\left(1+3 x \log _{e}(\cos y)\right) \sin \mathrm{y} d y=0\) satisfy $$x\left(\frac{\pi}{3}\rig...

Let \(y=y(x)\) be a solution of the differential equation \((x \cos x) d y+(x y \sin x+y \cos x-1) d x=0,0 < x < \frac{\pi}{2}\). If \(\frac{\pi}{3} y\left(\frac{\pi}{3}\right)=\sqrt{3}\), then $$\left|\frac{\pi}{6} y^{\prime \prime}\left(\...

If the solution curve \(f(x, y)=0\) of the differential equation
\(\left(1+\log _{e} x\right) \frac{d x}{d y}-x \log _{e} x=e^{y}, x > 0\),
passes through the points \((1,0)\) and \((\alpha, 2)\), then \(\alpha^{\alpha}\) is equal to :

Let a differentiable function \(f\) satisfy \(f(x)+\int_\limits{3}^{x} \frac{f(t)}{t} d t=\sqrt{x+1}, x \geq 3\). Then \(12 f(8)\) is equal to :

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

Let the solution curve \(y=y(x)\) of the differential equation $$
\frac{\mathrm{d} y}{\mathrm{~d} x}-\frac{3 x^{5} \tan ^{-1}\left(x^{3}\right)}{\left(1+x^{6}\right)^{3 / 2}} y=2 x \exp \left\{\frac{x^{3}-\tan ^{-1} x^{3}}{\sqrt{\left(1+x^{...

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

Let \(y=f(x)\) be the solution of the differential equation \(y(x+1)dx-x^2dy=0,y(1)=e\). Then \(\mathop {\lim }\limits_{x \to {0^ + }} f(x)\) is equal to

Let \(y=y(x)\) be the solution of the differential equation \(x{\log _e}x{{dy} \over {dx}} + y = {x^2}{\log _e}x,(x > 1)\). If \(y(2) = 2\), then \(y(e)\) is equal to

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

Let \(y=y(t)\) be a solution of the differential equation \({{dy} \over {dt}} + \alpha y = \gamma {e^{ - \beta t}}\) where, \(\alpha > 0,\beta > 0\) and \(\gamma > 0\). Then \(\mathop {\lim }\limits_{t \to \infty } y(t)\)

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

Let \(y=y(x)\) be the solution of the differential equation \((x^2-3y^2)dx+3xy~dy=0,y(1)=1\). Then \(6y^2(e)\) is equal to

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

The area enclosed by the closed curve \(\mathrm{C}\) given by the differential equation \(\frac{d y}{d x}+\frac{x+a}{y-2}=0, y(1)=0\) is \(4 \pi\).
Let \(P\) and \(Q\) be the points of intersection of the curve \(\mathrm{C}\) and the \(y\)-...

Let \(\alpha x=\exp \left(x^{\beta} y^{\gamma}\right)\) be the solution of the differential equation \(2 x^{2} y \mathrm{~d} y-\left(1-x y^{2}\right) \mathrm{d} x=0, x > 0,y(2)=\sqrt{\log _{e} 2}\). Then \(\alpha+\beta-\gamma\) equals :

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

$2(y+2) \log _{e}(y+2) d x+\left(x+4-2 \log _{e}(y+2)\right) d y=0, y>-1$

with $x\left(e^{4}-2\right)=1$. Then $x\left(e^{9}-2\right)$ is equal to :

Let \(y=y_{1}(x)\) and \(y=y_{2}(x)\) be the solution curves of the differential equation \(\frac{d y}{d x}=y+7\) with initial conditions \(y_{1}(0)=0\) and \(y_{2}(0)=1\) respectively. Then the curves \(y=y_{1}(x)\) and \(y=y_{2}(x)\) inte...

If \(y=y(x)\) is the solution of the differential equation \(\frac{d y}{d x}+\frac{4 x}{\left(x^{2}-1\right)} y=\frac{x+2}{\left(x^{2}-1\right)^{\frac{5}{2}}}, x > 1\) such that $$y(2)=\frac{2}{9} \log _{e}(2+\sqrt{3}) \text { and } y(\sqrt...

Let \(y=y(x), y > 0\), be a solution curve of the differential equation \(\left(1+x^{2}\right) \mathrm{d} y=y(x-y) \mathrm{d} x\). If \(y(0)=1\) and \(y(2 \sqrt{2})=\beta\), then