Question

solve the diffrential equation x dy/dx + y = x^2y^6

Answer 1

$x\frac{dy}{dx}+y=x^2y^6$

Solution:

Divide the left and right sides of the equation by $y^6$ :

$\frac{x}{y^6}\frac{dy}{dx}+\frac{1}{y^5}=x^2$

$\frac{1}{y^5}=z$ , $z'=-5\frac{y'}{y^6}$ , $\frac{y'}{y^6}=-\frac{z'}{5}$ .

$-\frac{xz'}{5}+z=x^2$

$xz'-5z=-5x^2$

$z=uv$ , $z'=u'v+v'u$

$xu'v+xv'u-5uv=-5x^2$

$xu'v+u(xv'-5v)=-5x^2$

Let's compose and solve the system:

$\begin{cases} xv'-5v=0 \\ xu'v=-5x^2 \end{cases}$

From the first equation:

$\frac{dv}{v}=\frac{5dx}{x}$

$\ln|v|=5\ln{|x|}$

$v=x^5$

Substitute $v$ into the second equation:

$xu'x^5=-5x^2$

$du=-5x^{-4}dx$

$u=\frac53x^{-3}+C$

$z=uv=x^5(\frac53x^{-3}+C)=\frac53x^2+Cx^5$

$\frac{1}{y^5}=\frac53x^2+Cx^5$

Answer: $\frac{1}{y^5}=\frac53x^2+Cx^5$ .