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andrezito [222]
3 years ago
8

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

Mathematics
1 answer:
Anni [7]3 years ago
6 0

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}.

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