Methods of Solving Differential Equations

Solve the differential equation $\frac{dy}{dx}=-\left[\frac{ x+y\cos x}{1+\sin x}. \right]$

The solution of $\frac{dy}{dx}=\frac{x^2-y^2+1}{2xy}$

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 $f\left(x\right)$ is the solution of the equation $\left(x+1\right)\frac{dy}{dx}-xy-1=0$ and $f\left(0\right)=0$, then $f\left(e\right)$ equals

The integrating factor of the differential equation $\cos y+\left(x\sin y-1\right)\frac{dy}{dx}=0$ 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

If the solution of the differential equation $xdy+2\sin y dy-\sec y dx=0$ is $x=c{e}^{\sin y}-k\left(1+\sin y\right)$, then the value of $k$ is equal to (where $c$ is the constant of integration)

If $y=y\left(x\right)$ and $x^2\frac{dy}{dx}+y=1$ satisfy $y\left(1\right)=1+e$, then the value of $y\left({\log }_{2}e\right)$ is 

If $y\left(x\right)$ is solution of $\frac{1}{y}\frac{dy}{dx}-\frac{\ln y}{x}=x^2$ and $y\left(\sqrt{2}\right)=1$, then $y\left(1\right)$ is equal to

Solution of the following differential equation

$\frac{dy}{dx}=x\sqrt{25-x^2}$ is

The solution of the following differential equation $y-x\frac{dy}{dx}=0$ is

Solution of the differential equation $\frac{dy}{dx}=\frac{x+y+1}{2x+2y+3}$ is

The particular solution of the differential equation: 

$y(1+\log x)\frac{dx}{dy}-x\log x=0$ when $y=e^2$ and $x=e$ is 

Find the general solution of the differential equation $x\frac{dy}{dx}+2y=x^2\log x$.

Integrating factor of the linear differential equation $x\frac{dy}{dx}+2y=x^2\log x$ is

The solution of $\frac{dy}{dx}=\frac{2y+x+1}{2x+4y+3}$

Selection of the differential equation

$\cos x\frac{dy}{dx}+y\sin x={\sec }^{2}x$

Solution of differential equation $\frac{dy}{dx}+y=e^{-x}$ is

Find the equation of a curve passing through the point $(2,3)$, given that the slope of the tangent to the curve at any point $(x,y)$ is $\frac{2x}{y^2}$.

Soultion of the differential equation $\left(1+y^2\right)+\left(x-e^{{\tan }^{-1}y}\right)\frac{dy}{dx}=0$ is