Calculus of variations, Euler Lagrange problem

Question

I have just started doing calculus of variations and have run into a wall trying to solve one of the first problems in solving the Euler-Lagrange equations.

$$\int_{0.1}^1 y'(1 + x^2y') \,dx, y(0.1) = 19, y(1) = 1$$

I have to find the Euler-Lagrange equation which satisfies the boundary conditions but every time I am not able to reach the actual solution.

Answer 1

Too long for a comment, I will be glad to later expand to a full answer. To find the Euler-Lagrange equation one has to compute $$ \frac{\partial \mathcal{L}}{\partial y} = \frac{\mathrm{d}}{\mathrm{d}x} \frac{\partial \mathcal{L}}{\partial y´}$$ where in your case $$ \mathcal{L} = y´(1+x^2y´) $$

Once you have obtained the Euler-Lagrange equation you can consider the boundary conditions. At what point of the process do you get stuck? How does your answer differ from the actual solution?

Answer 2

You do not have a $y$ term so E-L reduces to $\frac{d}{dx}\frac{\partial L}{\partial y'}=0$.

Giving $\frac{d}{dx} (1+2 x^2 y')=0$. This easily integrates to give $1+2 x^2 y'=const$. This rearranges to give $y'= const/x^2$. Can you take it from there?