Answer to Question #341203 in Differential Equations for Yes

Question

Solve x² dy/dx = y² - xy. given that y = 1 when x = 1

Answer 1

Solution

Dividing by x² we’ll get equation  dy/dx = y² / x²  – y/x

Substitution    z=y/x, y=z*x  =>  dy/dx = x(dz/dx) + z

From given equation  x(dz/dx)  = z² – 2z  =>  $\frac{dz}{z^2-2z}=\frac{dx}{x}$   =>  $\int\frac{dz}{z^2-2z}=\int\frac{dx}{x}$   =>   $\frac{1}{2}\int\left(\frac{1}{z-2}-\frac{1}{z}\right)dz=\int\frac{dx}{x}$   =>  ln|z–2| – ln|z| = 2ln|x| + C

If y = 1 when x = 1 then z = 1 when x = 1. Substitution this into last equality gives 0 – 0 = 2*0 + C  =>  C = 0.

So solution of the given equation is

ln|z–2| – ln|z| = 2ln|x|  => ln(|z–2| / |z|) = 2ln|x|  => |(y-2x)/y| = x² , 2x – y = y x² => y(1 + x²) = 2x => y(x) = 2x/(1 + x²)     

Answer

y(x) = 2x/(1 + x²)