Find the particular solution of the differential equation(adsbygoogle = window.adsbygoogle || []).push({});

[tex]xy' + 4y = -20xcos(x^5)[/tex]

satisfying the initial condition y(pi) = 0.

Solution

[tex]y' + 4(y/x) = -20cos(x^5)[/tex]

p(x) = [tex]4/x[/tex]

q(x) = [tex]-20cos(x^5)[/tex]

[tex]u(x) = int(4/x) = 4lnx[/tex]

[tex]e^{u(x)} = x^4[/tex]

[tex]1/e^{u(x)} int(e^{u(x)}*q(x)dx[/tex]

[tex]1/(x^4) int(x^4*(-20cos(x^5)))dx[/tex]

u substitution

[tex]u = x^5[/tex]

[tex]du = 5x^4dx[/tex]

[tex]1/(x^4) int(x^4*(-20cos(u)))(du/(5x^4))[/tex]

[tex]-4/(x^4) int(cosu)du[/tex]

[tex]-4/(x^4) (sin(x^5)+c)[/tex]

[tex](-4sin(x^5)/(x^4))-((4c)/(x^4))[/tex]

[tex]y(pi) = 0 = (-4sin(pi^5)/(pi^4))-((4c)/(pi^4))[/tex]

[tex]c = -sin(pi^5)[/tex]

Final

[tex](-4sin(x^5)/(x^4))+((4sin(pi^5))/(x^4))[/tex]

So this is what I got, but it's a wrong answer. Can anybody tell me what I do wrong in this problem?

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# Particular solutions - DE

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