## Euler-Legrange (Do the extremals envelope?)

1. The problem statement, all variables and given/known data

Calculate and sketch the extremals of function given by the integral between 1 and b of (x^2*y'^2+12y^2)

2. Relevant equations

The extremals pass through the point (1,1) and y(b) = v with b>1

3. The attempt at a solution
I have F(x,y,y') = (x^2)(y'^2) + 12y^2
Therefore the Euler legrange equation is;

24y - d/dx 2(x^2)(y') = 0 -> (x^2)y'' + 2xy' -12y = 0

Solving the ODE i get y = C1*x^-4 + C2*x^3

And using the condition that y(1) = 1 and b>1 i have sketched a few suitable extremals.

However when it asks if the extremals envelope i'm not sure what that means?

Any help much appreciated! As an aside what is the difference between an extremum and extremal?
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