Question

The indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula below:

y2 = y1(x) ∫ [e^(−∫P(x) dx)/ y²₁(x)]dx
as instructed, to find a second solution y2(x).
y'' − 4y' + 4y = 0; y1 = e^(2x)
y2 =?

Answer 1

Given
`y_1=e^(2x)` is a fundamental solution, we posit a second solution of the form
`y_2=y_1v=e^(2x)v`, with derivatives

`{y_2}'=e^(2x)v'+2e^(2x)v=e^(2x)(v'+2v)`

`{y_2}''=e^(2x)v''+4e^(2x)v'+4e^(2x)v=e^(2x)(v''+4v'+4v)`

Substitute these into the ODE:

`e^(2x)(v''+4v'+4v)-4e^(2x)(v'+2v)+4e^(2x)v=0\implies v''=0`

Integrate both sides twice to get

`v''=0\implies v'=C_1\implies v=C_1x+C_2`

Then the second fundamental solution is

`y_2=xe^(2x)+e^(2x)`

but
`y_1` already cover
`e^(2x)`, so
`y_2=xe^(2x)`.