# Simplify. x(dy/dx) - 3y - x^6 e^(x^3) , y(0) = 2

Simplify. $x\left(dy/dx\right)-3y-{x}^{6}e{x}^{3},y\left(0\right)=2$
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Jazlene Dickson
$\frac{dy}{dx}-\frac{3y}{x}={x}^{5}{e}^{{x}^{3}}$ linear diff. eqn.
y' +P(x) y = Q(x)
P(x)=-3/y
$Q\left(x\right)={x}^{5}\mathrm{exp}\left({x}^{3}\right)$
$i\left(x\right)=\mathrm{exp}\left(\int \frac{-3}{x}dx\right)=\mathrm{exp}\left(-3\mathrm{ln}x\right)=\mathrm{exp}\left(\mathrm{ln}{x}^{-3}\right)=\frac{1}{{x}^{3}}$
$y=\int \frac{1}{{x}^{3}}{x}^{5}{e}^{{x}^{3}}dx+c$
$y=\int {x}^{2}{e}^{{x}^{3}}dx+c$
${x}^{3}=u$
$3{x}^{2}dx=du$
${x}^{2}dx=du/3$
$y=\frac{1}{3}\int {e}^{u}du+c$
$y=\frac{{e}^{{x}^{3}}}{3}+c$
$y=\frac{{x}^{3}{e}^{{x}^{3}}}{3}+c{x}^{3}$