Find the equation of the curve through the origin, which satisfies the differential equation $\frac{dy}{dx}=2x\left(y^2+9\right)$.

Find the solution of the following differential equation:

${\left(x+y\right)}^2\frac{dy}{dx}=a^2$

Find the solution of the following differential equation :

$\frac{dy}{dx}={\left(y-x\right)}^2$

Find the solution of the following differential equation :

$\frac{dy}{dx}=\sin \left(x+y\right)$

Find the solution of the following differential equation :

$\frac{dy}{dx}={\sec }^2\left(x+y\right)$

Find the solution of the following differential equation :

$\frac{dy}{dx}=\sin \left(x+y\right)+\cos \left(x+y\right)$

Find the solution of the following differential equation :

$\frac{dy}{dx}+1=e^{x-y}$

Find the solution of the following differential equation :

$\frac{dy}{dx}-x\tan \left(y-x\right)=1$

Find the solution of the following differential equation :

$\frac{dy}{dx}=\sqrt{y-x}$