Variational Problem.

Question

How to solve variational problem $$I(y)=\int_0^1 [(y’)^2-y|y|y’+xy]dx,y(0)=y(1)=0?$$

I tried by Euler equation, which is $$ -2|y|y’+x-\frac{d}{dx}(2y’-y|y|)=0$$ Now stuck. Unable to creat corresponding differential equation. Please help. Thank you.

Answer 1

Hint: \begin{align} \frac{d}{dx}(y|y|)&=\frac{d}{dx} \begin{cases} y^2&\text{if $y\geq 0$,} \\ -y^2&\text{if $y<0$,} \end{cases} \\ &=\begin{cases} 2yy'&\text{if $y\geq 0$,} \\ -2yy'&\text{if $y<0$,} \end{cases} \\ &=2|y|y', \tag{1} \end{align} hence $$ -2|y|y’+x-\frac{d}{dx}(2y’-y|y|)=0 \implies x-2y''=0. \tag{2} $$