Determine the order and the degree (when defined) of differential equation, $y=px+\sqrt{a^2{p}^2+b^2}$ Where $p=\frac{dy}{dx}$.

Write the sum of the order and the degree of the differential equation: ${\left(\frac{d^{2}y}{d{x}^2}\right)}^2-{\left(\frac{dy}{dx}\right)}^3=y^3$.

Find the product of the order and degree of the differential equation, $x{\left(\frac{d^{2}y}{d{x}^2}\right)}^2+{\left(\frac{dy}{dx}\right)}^2+y^2=0$.

What is the number of arbitrary constants in the general solution of a differential equation of order $4$?

What is the number of arbitrary constants in a particular solution of a differential equation of order $3$?

What is the number of arbitrary constants in the general solution of the differential equation $\left(1+x^2\right){\left(\frac{d^{2}y}{d{x}^2}\right)}^3+y{\left(\frac{dy}{dx}\right)}^4=7\sqrt{x} ?$

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

Show that the given function $y=\cos x+C$ is a solution of the differential equation $\frac{dy}{dx}+\sin x=0$.

The function $y=x^2+2x+C$ is a solution of differential equation $y\text{'}-ax-b=0$. Find the value of $a$ and $b$.