Answer to Question #339548 in Differential Equations for Themba

Question

Solve the following differential equations: (x^3+y^3)=(xy^2)dy/dx

Answer 1

Solution

Given differential equation may be rewritten in the form

x³ + y³ = xy²dy/dx  =>  x²/y² + y/x = dy/dx  

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

From this and reduced equation

(dz/dx)x + z = 1/z² + z  =>  dz/dx = 1/(xz²)  =>  z²dz = dx/x  =>

$\int{z^2dz}=\int\frac{dx}{x}$ =>   $\frac{1}{3}z^3=ln\left|x\right|-C$   =>  (y/x)³ = 3ln|x| - 3C  =>  

$y(x)=x\sqrt[3]{3(ln\left|x\right|-C)}$

Answer

$y(x)=x\sqrt[3]{3(ln\left|x\right|-C)}$