Question

Solve the differential equation (x³ + y³)dy - x²y dx = 0.

Answer 1

(x³ + y³)dy - x²y dx = 0

dy/dx = (y³ + x³)/x²y ...(i)

This is a homogenous equation so, we put y = vx and dy/dx = v + x(dv/dx)

Hence, (i) becomes

Now Integrating, we get

log|x| + log(1 + v³) = log c

log[|x|(1 + v³)] = log c

|x|[1 + (y³/x³)] = c

x³ + y³ = c|x| = ±cx 

= kx (let)