How resolve Lagrange equation ?

Question

How I can resolve this taks? I complety green in differential equations $y = xy' +y'^5$

Answer 1

Since you ar complety green in differential equations, I suppose that the solution could be simple such as polynomials.

Considering the equation , suppose that $y$ is a polynomial of degree $n$. So, $y'$ is a polynomial of degree $n-1$.

Comparing the degrees of lhs and rhs, we would have $$n=[1+(n-1)]+5\times(n-1)=6n-5$$ So,$\cdots ???$