Question

X^4y''=(y')^2+3x^6 solve differential equation

Answer 1

first put
u=y′/x²
y′′=2xu+x²u′ =u² + 3x²
x²u′=u²−2xu+x²+2x² = (u - x)² 2x² 
u' = ((u - x)/(x))²  +  2
now put
v=(u−x)/x
u=vx+x
u' = xv' + v +  1
xv′+v+1=v²+2
xv′=v²−v+1
dv/(v² - v +1) = dx/(x)

now use normal differential solution formula to find out the solution 
I hope it helps to you