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

Leila Jennings
2022-08-01
Answered

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

You can still ask an expert for help

Jazlene Dickson

Answered 2022-08-02
Author has **15** answers

$\frac{dy}{dx}-\frac{3y}{x}={x}^{5}{e}^{{x}^{3}}$ linear diff. eqn.

y' +P(x) y = Q(x)

P(x)=-3/y

$Q(x)={x}^{5}\mathrm{exp}({x}^{3})$

$i(x)=\mathrm{exp}(\int \frac{-3}{x}dx)=\mathrm{exp}(-3\mathrm{ln}x)=\mathrm{exp}(\mathrm{ln}{x}^{-3})=\frac{1}{{x}^{3}}$

$y=\int \frac{1}{{x}^{3}}{x}^{5}{e}^{{x}^{3}}dx+c$

$y=\int {x}^{2}{e}^{{x}^{3}}dx+c$

${x}^{3}=u$

$3{x}^{2}dx=du$

${x}^{2}dx=du/3$

$y=\frac{1}{3}\int {e}^{u}du+c$

$y=\frac{{e}^{{x}^{3}}}{3}+c$

$y=\frac{{x}^{3}{e}^{{x}^{3}}}{3}+c{x}^{3}$

y' +P(x) y = Q(x)

P(x)=-3/y

$Q(x)={x}^{5}\mathrm{exp}({x}^{3})$

$i(x)=\mathrm{exp}(\int \frac{-3}{x}dx)=\mathrm{exp}(-3\mathrm{ln}x)=\mathrm{exp}(\mathrm{ln}{x}^{-3})=\frac{1}{{x}^{3}}$

$y=\int \frac{1}{{x}^{3}}{x}^{5}{e}^{{x}^{3}}dx+c$

$y=\int {x}^{2}{e}^{{x}^{3}}dx+c$

${x}^{3}=u$

$3{x}^{2}dx=du$

${x}^{2}dx=du/3$

$y=\frac{1}{3}\int {e}^{u}du+c$

$y=\frac{{e}^{{x}^{3}}}{3}+c$

$y=\frac{{x}^{3}{e}^{{x}^{3}}}{3}+c{x}^{3}$

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