Verify that the function $y=e^{-3x}$is a solution of the differential equation $\frac{d^{2}y}{{dx}^2}+\frac{dy}{dx}-6y=0$

Find the general solution of the differential equation $\frac{dy}{dx}-y=\cos x$

The differential equations, find a particular solution satisfying the given condition: $\cos \left(\frac{dy}{dx}\right)=a\left(a\in R\right);y=1$

Verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.(i) $y=a{e}^x+b{e}^{-x}+x^2$ : $x\frac{d^{2}y}{{dx}^2}+2y\frac{dy}{dx}-xy+x^2-2=0$

Show that the given differential equation is homogeneous and solve each of them. $\left(x–y\right)dy–\left(x+y\right)dx=0$

Form the differential equation of the family of circles touching the y-axis at origin.

Solve the differential equation $\left({\tan }^{-1}y-x\right)dy=\left(1+y^2\right)dx$.

Find the general solution of the differential equations $\frac{dy}{dx}=\left(1+x^2\right)\left(1+y^2\right)$