The solution of $x\frac{dy}{dx}=y+x{e}^{\frac{y}{x}}$ with $y\left(1\right)=0$, is

Important Questions on Differential Equations

HARD

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

The solution of the differential equation $\frac{dy}{dx}=\frac{2xy-y^2}{2xy-x^2},$ is

HARD

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

The equation of the curve which passes through point $(1,0)$ and has tangent with the slope $1+\frac{y}{x}+{\left(\frac{y}{x}\right)}^2$ is

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

The general solution of the differential equation $\left(1+e^{\frac{x}{y}}\right)dx+\left(1-\frac{x}{y}\right)e^{x/y}dy=0$ is
$(C$ is an arbitrary constant$)$

HARD

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

Let $y=y\left(x\right)$ be the solution of the differential equation $x \tan \left(\frac{y}{x}\right)dy=\left(y \tan \left(\frac{y}{x}\right)-x\right)dx$, $-1\le x\le 1,y\left(\frac{1}{2}\right)=\frac{\pi }{6}.$ Then the area of the region bounded by the curves $x=0,x=\frac{1}{\sqrt{2}}$ and $y=y\left(x\right)$ in the upper half plane is:

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

The solution of $\frac{dy}{dx}=\frac{y^2}{xy-x^2}$ is

HARD

Mathematics>Integral Calculus>Differential Equations>Homogeneous 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 through $\left(1,1\right),$ then the area (in sq. units) enclosed by the curves $C_1$ and $C_2$ is equal to :

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

If $x\sin \left(\frac{y}{x}\right)dy=\left[y\sin \left(\frac{y}{x}\right)-x\right]dx,x>0$ and $y\left(1\right)=\frac{\pi }{2},$ then the value of $\cos \left(\frac{y}{x}\right)$ is

HARD

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

Solution of the differential equation $x\frac{dy}{dx}=y-x\tan \left(\frac{y}{x}\right)$ will be

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

The general solution of the differential equation $\frac{dy}{dx}=\frac{x+y-3}{x+y-7}$ is

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

The general solution of the differential equation $\left(x^2+xy\right)y^{\text{'}}=y^2$ is

HARD

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

If $y\frac{\mathrm{d}y}{ \mathrm{d}x}=x\left[\frac{y^2}{x^2}+\frac{\varphi \left(\frac{y^2}{x^2}\right)}{{\varphi }^{\text{'}}\left(\frac{y^2}{x^2}\right)}\right], x>0, \varphi >0,$ and $y\left(1\right)=-1,$ then $\varphi \left(\frac{y^2}{4}\right)$ is equal to:

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

If $\frac{dy}{dx}=\frac{xy}{x^2+y^2};y\left(1\right)=1;$ then a value of $x$ satisfying $y\left(x\right)=e$ is:

HARD

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

The solution of the differential equation $\frac{dy}{dx}=\frac{xy}{x^2+y^2}$ is

HARD

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

If $\frac{dy}{dx}=\frac{y+x \tan \frac{y}{x}}{x}$, then $\sin \frac{y}{x}=$

HARD

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

The solution of the differential equation $x\frac{dy}{dx}=y\left({\log }_{e}y-{\log }_{e}x+1\right)$ will be

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

The general solution of the differential equation $\left(x^2+y^2\right)\mathrm{d}x-2xy\mathrm{d}y=0$ is

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

A solution of the differential equation $2xydy-\left(x^2+y^2\right)dx=0$ satisfying $y\left(1\right)=1$ is given by

HARD

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

The solution of the differential equation $x^2\frac{dy}{dx}=y^2+xy$ is

HARD

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

The general solution of $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{2\mathrm{x}-\mathrm{y}}{\mathrm{x}+2\mathrm{y}}$ is given by

HARD

Mathematics>Integral Calculus>Differential Equations>Homogeneous Differential Equations

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