(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For the functional [tex]J(y(x))=\int^{x1}_{x2}F(x,y,y')dx[/tex], write out the curve [tex]y=y(x)[/tex] for finding the extremas of J where [tex]F(x,y,y')=y'^2+y^2[/tex].

2. Relevant equations

Euler's Equations:

[tex]\frac{\partial f}{\partial y} - \frac{d}{dx}\frac{\partial f}{\partial y'}=0[/tex]

[tex]\frac{\partial f}{\partial x} - \frac{d}{dx}(f-y' \frac{\partial f}{\partial y'})=0[/tex]

3. The attempt at a solution

Using [tex]\frac{\partial f}{\partial y} - \frac{d}{dx}\frac{\partial f}{\partial y'}=0[/tex],

[tex]\frac{\partial f}{\partial y}=2y[/tex]

[tex]2y=\frac{d}{dx}\frac{\partial f}{\partial y'}[/tex]

[tex]2y=\frac{d}{dx}2y'[/tex]

[tex]y=\frac{d^2y}{dx^2}[/tex]

[tex]y=C*e^x[/tex] Where C is a constant.

Is this correct? Using the 2nd equation, I get an ugly answer that involves Sinh.

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# Homework Help: Euler's Equation

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