xy^2*(dy/dx)=y^3-x^3, y(1)=2 (its a initial value problem) Any kind soul here who can give the solution of this eqn?

1 Answer
Dec 16, 2017

y = x root(3)(1/3(C_0-log_ex))

Explanation:

Making the variable change

y = lambda x rArr y' = x lambda'+lambda we have

x^3lambda^2(x lambda'+lambda)=x^3lambda^3-x^3 or

lambda' lambda^2x+1=0

This is a separable differential equation. Grouping variables

lambda^2d lambda=-dx/x and integrating

1/3lambda^3=-log_e x + C_0 or

lambda = root(3)(1/3(C_0-log_ex)) or

y = x root(3)(1/3(C_0-log_ex))

but y(1)=2=1 xx root(3)(C_0/3) rArr C_0 = 24