Question

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

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 $y_2=xe^{2x}$.