CoV9

From Exampleproblems

Find the extremals of $\int_a^b (y^2 +y'^2 + 2y e^x) \,dx\,$ .

The Euler-Lagrange equations are in general for the integrand functional $L\,$

$\frac{\partial L}{\partial y} - \frac{d}{dx}\frac{\partial L}{\partial y'} + \frac{d^2}{dx^2}\frac{\partial L}{\partial y''} - \frac{d^3}{dx^3}\frac{\partial L}{\partial y'''} +...+(-1)^k\frac{d^k}{dx^k}\frac{\partial L}{\partial y^{(k)}}=0\,$ .

So in this problem the EL-equation is

$\frac{\partial L}{\partial y} - \frac{d}{dx}\frac{\partial L}{\partial y'}=0\,$ .

$2y + 2e^x - \frac{d}{dx} 2y' =0\,$

$2y + 2e^x - 2y'' = 0\,$

$y'' - y = e^x\,$

The characteristic equation is $m^2-1=0, m=\pm 1\,$

$y_h = c_1 e^x + c_2 e^{-x}\,$

$y_p = A x e^x\,$

$y_p' = A x e^x + A e^x\,$

$y_p'' = A x e^x + A e^x + A e^x\,$

$y_p'' - y_p = 2A e^x = e^x, A=\frac{1}{2}\,$

$y_p = \frac{1}{2} x e^x\,$

$y(x) = c_1 e^x + c_2 e^{-x} + \frac{1}{2} x e^x\,$

Main Page : Calculus of Variations

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