Differential Equations

Let $C_{1}$ be the curve obtained by the solution of differential equation $2xy\frac{dy}{dx}=y^{2}-x^{2},x>0$.Let the curve $C_{2}$ be the solution of$\frac{2xy}{x^{2}-y^{2}}=\frac{dy}{dx}$. If both the curves pass…

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

If the differential equation representing the family of all circles touching $x$-axis at the origin is $\left(x^{2}-y^{2}\right)\frac{dy}{dx}=g\left(x\right)y$, then $g\left(x\right)$ equals

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

Suppose$y=y\left(x\right)$be the solution curve to the differential equation$\frac{dy}{dx}-y=2-e^{-x}$such that$\underset{x\rightarrow\,\infty\,}{\lim\,}y\left(x\right)$is finite. If$a$and$b$are respectively…

If$\frac{dy}{dx}+e^{x}\left(x^{2}-2\right)y=\left(x^{2}-2x\right)\left(x^{2}-2\right)e^{2x}$and$y\left(0\right)=0$,thenthe value of$y\left(2\right)$is

Let $y=y(x)$ be the solution of the differentialequation $xdy-ydx=\sqrt{\left(x^{2}-y^{2}\right)}dx,x\geq\,1$, with$y(1)=0$. If the area bounded by the line $x=1,x=e^{\pi\,},y=0$and $y=y(x)$ is…

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

If$y=y\left(x\right)$ is the solution of the differential equation$\frac{5+e^{x}}{2+y}\cdot\,\frac{dy}{dx}+e^{x}=0$ satisfying$y\left(0\right)=1$thenvalue of$y(\log\,_{e}13)$is

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

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

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