Question

Solve the following differential equations: `2x^(2)y (d^(2)y)/(dx^(2))+4y^(2)=x^(2)(dy/dx)^(2)+2xy(dy/dx)`?

Answer 1

# y = x^2(C_1+C_2 ln x)^2 #

Explanation:

Making the change of variables `y(x) = x^2 z^2` and substituting into the DE we get

# 4x^5 z^3(z'+x z'') = 0 #

so assuming `z \ne 0` we have

# z'+xz'' = 0\to z = C_1+C_2 ln x #

and then

# y = x^2(C_1+C_2 ln x)^2 #