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

Solve the linear differential equation:

[tex] xy'-2y=x^{2} [/tex]

2. Relevant equations

If you have a linear differential equation of the form:

[tex] y'+P(x)y=Q(x) [/tex]

then your integrating factor is:

[tex] I(x)=e^{\int P(x) dx} [/tex]

3. The attempt at a solution

If we divide both sides by x then the equation is in standard form:

[tex] y' - \frac {2y}{x} = x [/tex]

where

[tex] P(x)=-\frac{2}{x} [/tex]

thus:

[tex] I(x)=e^{\int -\frac{2}{x} dx} = e^{-2\int\frac{1}{x}dx}= e^{-2ln|x|}=x^{-2} [/tex]

so we then multiple both sides of the diff eq by I(x):

[tex] x^{-2}y'-2x^{-3}y=\frac{1}{x} [/tex]

which is:

[tex] \frac {d}{dx}(x^{-2}y)=\frac{1}{x} [/tex]

if we the integrate both sides:

[tex] x^{-2}y=ln|x|+c [/tex]

therefore:

[tex] y= x^{2}ln|x|+cx^{2} [/tex]

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# Homework Help: Linear diff eq - correctly done?

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