Consider the variational problem (P)

J(y(x)) = [(y')² − y|y| y' + xy] dx, y(0) = 0, y(1) = 0.

Which of the following statements is correct?

Concept:

Euler-Lagrange Equation: The extremal of the functional J[y] = f(x, y, y')dx satisfies  - () = 0

Explanation:

J(y(x)) = [(y')2 − y|y| y' + xy] dx, y(0) = 0, y(1) = 0

If y > 0 then

f(x, y, y') = (y')2 − y²y' + xy

So using   - () = 0 we get

-2yy' + x - (2y' - y²) = 0

- 2yy' + x - 2y'' +2yy' = 0

y'' = x/2....(i)

If y < 0 then

f(x, y, y') = (y')2 + y2y' + xy

So using   - () = 0 we get

2yy' + x - (2y' + y2) = 0

 2yy' + x - 2y'' - 2yy' = 0

y'' = x/2....(ii)

Hence in both case we get

y'' = x/2

Integrating 

y' = 

Integrating again

y = 

Using  y(0) = 0, y(1) = 0 we get

c₂ = 0 and 0 =  + c₁ ⇒ c1 = - 

Hence solution is

y = 

Option (3) is correct