Exact Differential Equations

The general solution of the equation $e^y\left(\frac{dy}{dx}+1\right)=e^x$ is

The general solution of the differential equation $\left(x^2{y}^2+y\right)dx-\left(x-2x^{3}y\right)dy=0$ is

Let $y_1\left(x\right)$ and $y_2\left(x\right)$ are the solutions of the differential equation $\left(y\cos x-e^x\cos x\right)dx=\left(dy-e^{x}dx\right)\sin x$ and $y_1\left(x\right)$ passes through the point $\left(\frac{\pi }{2},{\mathrm{e}}^{\frac{\pi }{2}}\right)$ and $y_2\left(x\right)$ passes through the point $\left(\frac{-\pi }{2},1\right)$, then which of the following is/are true?

Solution of differential equation $f\left(x\right)\frac{dy}{dx}=f^2\left(x\right)+f\left(x\right)y+f^{\text{'}}\left(x\right)y$ is

General solution of the differential equation $x^4\frac{dy}{dx}+x^{3}y+cosec\left(xy\right)=0$ is 

Solve the following differential equation

$x^{2}y\left(2xdy+3ydx\right)=dy$

Solve:

$4\frac{dy}{dx}=\frac{\sqrt{3}x-4y+7}{x-y}$

Solve:

$\left(2x-y+1\right)dx+\left(2y-x-1\right)dy=0$

The solution of the differential equation $\frac{ydx+xdy}{ydx-xdy}=\frac{x^2{e}^{xy}}{y^4}$ satisfying $y\left(0\right)=1$, is

If for the differential equation $ydx+y^{2}dy=xdy, x\in R,y>0$ and  $y\left(1\right)=1$ , then $y \left(-3\right)$ is equal to:

Curve satisfying the primitive integral equation $y^{2}dy=xdy-ydx, x\in R, y\in R^{+}$ passes through $\left(4,2\right)$, then the $y$ coordinate of the point on this curve having $x$ coordinate $-96$ is

If the solution of the differential equation $xydx+x^{2}dy+e^{-xy}dx=x^3{e}^{-xy}dx$ satisfy $y\left(1\right)=0$, then $y\left(e\right)$ is equal to

Let $a$ differential equation be $\frac{dy}{dx}=\frac{x^2+y^2+1}{2xy}$. Then the general solution of this differential equation is

The solution of the differential equation $\frac{\mathrm{ydx}-\mathrm{xdy}}{\mathrm{xy}}=\mathrm{xdx}+\mathrm{ydy}$ is (where, $\mathrm{C}$ is an arbitrary constant)

If $x dy=y(d x+y dy), y\left(1\right)=1$ and $y\left(x\right)>0 .$ Then, $y (-3)$ is equal to

Let $f\left(x\right)$ be differentiable on the interval $\left(0,\infty \right)$ such that $f\left(1\right)=1$, and $\underset{t\to x}{\lim }\frac{t^{2}f\left(x\right)-x^{2}f\left(t\right)}{t-x}=1$ for each $x>0$. Then, $f\left(x\right)$ is

The general solution of the differential equation $\left(y^2-x^3\right)dx-xydy=0 (x\ne 0)$ is (where, $C$ is a constant of integration)

Find solution of the differential equation $\frac{xdy}{ydx}+\frac{\sin \left(y\right)+\cos \left(x\right)\ln y^x}{\sin \left(x\right)+\cos \left(y\right)\ln x^y}=0$.

Find the solution of the differential equation $\frac{xdy}{x^2+y^2}=\left(\frac{y}{x^2+y^2}-1\right)dx$ 

Differential equation $\left\{\frac{1}{x}-\frac{y^2}{{\left(x-y\right)}^2}\right\}dx+\left\{\frac{x^2}{{\left(x-y\right)}^2}-\frac{1}{y}\right\}dy=0$ has the solution _________ {$c$ is an arbitrary constant}