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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 :

LammettHash

Given [tex]y_1=e^{2x}[/tex] is a fundamental solution, we posit a second solution of the form [tex]y_2=y_1v=e^{2x}v[/tex], with derivatives

[tex]{y_2}'=e^{2x}v'+2e^{2x}v=e^{2x}(v'+2v)[/tex]

[tex]{y_2}''=e^{2x}v''+4e^{2x}v'+4e^{2x}v=e^{2x}(v''+4v'+4v)[/tex]

Substitute these into the ODE:

[tex]e^{2x}(v''+4v'+4v)-4e^{2x}(v'+2v)+4e^{2x}v=0\implies v''=0[/tex]

Integrate both sides twice to get

[tex]v''=0\implies v'=C_1\implies v=C_1x+C_2[/tex]

Then the second fundamental solution is

[tex]y_2=xe^{2x}+e^{2x}[/tex]

but [tex]y_1[/tex] already cover [tex]e^{2x}[/tex], so [tex]y_2=xe^{2x}[/tex].

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