Question

Solve:

$\frac{d^{2}y}{d{x}^2}+3\frac{dy}{dx}-54y=0$.

Answer 1

Given the differential equation,

$\frac{d^{2}y}{d{x}^2}+3\frac{dy}{dx}-54y=0$.......(1).

Let $y=c{e}^{mx}$ for $c\ne 0$ be a trial solution of equation (1).

Then the auxiliary equation is

$m^2+3m-54=0$

or, $m^2+9m-6m-54=0$

or, $(m+9)(m-6)=0$.

or, $m=6,-9$.

So the solution is $y=c_1{e}^{-9x}+c_2{e}^{6x}$. [ Where $c_1$ and $c_2$ are arbitrary constant.]