Differential Equations

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

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

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 $\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{2 x}{\left(1+x^2\right)^2} y=x \mathrm{e}^{\frac{1}{\left(1+x^2\right)}} ; y(0)=0 .$ Then the area enclosed by the…

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 :

Let the tangent at any point $P$on a curve passing through the points $\left(1,1\right)$and $\left(\frac{1}{10},100\right)$, intersect positive $x$-axis and $y$-axis at the points $A$ and $B$ respectively. If…

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

Let $y = y(x)$ be the solution of the differential equation $x\sin\left(\dfrac{y}{x}\right)dy = \left(y\sin\left(\dfrac{y}{x}\right) - x\right)dx$, $y(1) = \dfrac{\pi}{2}$ and let $\alpha =…

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…

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

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 ________