Homogeneous Differential Equation

Question

For what value of n is the homogeneous differential equation:

                              \(dy\over dx\)\(= {{x^3-y^n} \over x^2y+xy^2}\)

Answer 1

Given differential equation is 2x4ydxdy​+y4=4x4,

Substituting y=un

⟹dxdy​=nun−1dxdu​ gives,

⟹2x4un(nun−1dxdu​)=4x4−u4n⟹dxdu​=nu2n−12​−2nx4u2n+1​,

Given this equation is homogenous,

By inspection,if n=21​, 

The equation becomes dxdu​+x4u2​=4, which is homogenous.

Comments

Wrong Answer because you solve other differential equation. I didn't give the equation that you wrote above.