cistG

2020-12-30

Solve differential equation $x{y}^{\prime }+3y=6{x}^{3}$

Usamah Prosser

$⇒x\frac{dy}{dx}+3y=6{x}^{3}$
Divide by 'x'
$⇒\frac{x\frac{dy}{dx}}{x}+\frac{3y}{x}=\frac{6{x}^{3}}{x}$
$\int \frac{dy}{dx}+\frac{3}{x}y=6{x}^{2}$ (1)
Compare (1) with $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$ we have
$P\left(x\right)=\frac{3}{x}$, $Q\left(x\right)=6{x}^{2}$
$I.F.={e}^{\int P\left(x\right)dx}$
$={e}^{3\int \frac{1}{x}dx}$
$={e}^{3\mathrm{ln}\left(x\right)}$ ($\because \int \frac{1}{x}=\mathrm{ln}\left(x\right)$)
$I.F.={e}^{\mathrm{ln}{x}^{3}}$ ($\because n\mathrm{ln}x=\mathrm{ln}{x}^{n}$)
$I.F.={x}^{3}$
$y\left(I.F.\right)=\int Q\left(x\right)\left(I.F.\right)dx$
$\int y{x}^{3}=\int 6{x}^{2}{x}^{3}dx$
$\int {x}^{3}y=6\int {x}^{5}dx$ ($\because {a}^{m}{a}^{n}={a}^{m+n}$)
$\int {a}^{n}dx=\frac{{a}^{n+1}}{n+1}$
$⇒{x}^{3}y=6\left(\frac{{x}^{5+1}}{5+1}\right)+C$
$⇒{x}^{3}y=6\left(\frac{{x}^{6}}{6}\right)+C$
$⇒{x}^{3}y={x}^{6}+C$
$⇒y=\frac{{x}^{6}}{{x}^{3}}+\frac{C}{{x}^{3}}$
$⇒y={x}^{3}+\frac{C}{{x}^{3}}$

Jeffrey Jordon