Question

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

Answer 1

The general solution:
`C_(1)e^(-4x) + xC_(2)e^(-4x)`

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)`