(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

You are given that two solutions to the homogeneous Euler-Cauchy equation

[itex]x^2 \frac{d^2}{dx^2}y(x) - 5x \frac{d}{dx} y(x) + 5y(x) = 0[/itex]

[itex]y1=x, y2=x^5[/itex]

[itex]y''-\frac{5}{x}y'+\frac{5}{x^2}y=-\frac{49}{x^4}[/itex]

changing the equation to standard form

use variation of parameters to find a particular solution to the inhomogenous Euler-Cauchy equation

2. Relevant equations

Wronskian

[itex]W=4x^5 [/itex]

yp (y particular)

[tex]yp=uy1+vy2 [/tex]

[tex]u= \frac{-49}{12x^3} [/tex]

3. The attempt at a solution

[itex]v' = \frac{y1r}{w} [/itex]

[itex]v' = \frac{(x) (-49/x^4)}{4x^5} [/itex]

[itex]v' = -\frac{49}{4x^8}[/itex]

[itex]v = -\frac{49}{4} \int \frac{1}{x^8} [/itex]

[itex]v = (-\frac{49}{4}) (-\frac{1}{7x^7}) [/itex]

[itex]v = \frac{49}{28x^7}[/itex]

[itex]yp = \frac{-49}{12x^3}*x + \frac{49}{28x^7}*x^5[/itex]

[itex]yp = \frac{-49}{12x^2} + \frac{49}{28x^2}[/itex]

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# Homework Help: ODE using variation of parameters

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