# xy'−3y=x−1? Solve

$xy\prime -3y=x-1$ Solve
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Griffin Dean
Divide all terms by x
$\frac{dy}{dx}-\frac{3y}{x}=\frac{x-1}{x}$
The ODE is now in a form where it can be solved using an integrating factor
Let $\mu =\mathrm{exp}\int -\frac{3}{x}=\mathrm{exp}\left(-3\mathrm{ln}x\right)=\mathrm{exp}\mathrm{ln}\left(\frac{1}{{x}^{3}}\right)=\frac{1}{{x}^{3}}$
Now multiply the whole equation by $\mu$
$\frac{1}{{x}^{3}}\frac{dy}{dx}-\frac{3y}{{x}^{4}}=\frac{x-1}{{x}^{4}}$
The LHS factors as a derivative and we can integrate the RHS
$\int \frac{d}{dx}\left(\frac{y}{{x}^{3}}\right)dx=\int {x}^{-3}-{x}^{-4}dx$
$\frac{y}{{x}^{3}}=-\frac{1}{2}{x}^{-2}+\frac{1}{3}{x}^{-3}+a$
Solve for y
$y=y\left(x\right)=a{x}^{3}-\frac{1}{2}x+\frac{1}{3}$

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