Question

Find a general solution of y" + 8y' + 16y=0.

Answer 1

Answer:

The general solution: $C_{1}e^{-4x} + xC_{2}e^{-4x}$

Step-by-step explanation:

Differential equation: y'' + 8y' + 16y = 0

We have to find the general solution of the above differential equation.

The auxiliary equation for the above equation can be writtwn as:

m² + 8m +16 = 0

We solve the above equation for m.

(m+4)² = 0

$m_{1}$ = -4, $m_{2}$ = -4

Thus we have repeated roots for the auxiliary equation.

Thus, the general solution will be given by:

y = $C_{1}e^{m_{1}x} + xC_{2}e^{m_{2}x}$

y = $C_{1}e^{-4x} + xC_{2}e^{-4x}$