Eliminate the arbitrary constants and obtain the differential equation satisfied by it.

Question

\(y=ae^{2x}+be^{-2x}+c\)

Answer 1

\(y = ae^{2x} + be^{-2x} + c\)

\(\frac{dy}{dx} = 2ae^{2x} - 2be^{-2x}\)

\(\frac{d^2y}{dx^2} = 4ae^{2x} + 4be^{-2x}\)

\(= 4(ae^{2x} + be^{-2x})\)

\(= 4(y - c)\)

\(= 4y - 4c\)

\(\therefore \frac{d^3y}{dx^3} = 4\frac{dy}{dx}\)

⇒ \(\frac{d^3y}{dx^3} - 4\frac{dy}{dx}=0\)

which is required differential equation.