Recent content by mizzcriss

To start, thanks for your help! :-) The textbook used for the class doesn't cover calculus of variation, my professor covered it separately and only derived the simple case. I have tried to apply the simple case directly to my second problem but I don't know what to do with the ##y'## and...

Homework Statement Problem 1: Derive the Euler-Lagrange equation for the function ##z=z(x,y)## that minimizes the functional $$J(z)=\int \int _\Omega F(x,y,z,z_x,z_y)dxdy$$ Problem 2: Derive the Euler-Lagrange equation for the function ##y=y(x)## that minimizes the functional...

Wow thank you so so much! :smile: From the 2 equations I came up with $$F(x)=f(x)+H(x)$$ and $$G(x)=f(x)-H(x)$$ Then adding them gives me essentially exactly what I need but in the problem the solution ##u(x,t)=f(x+t)+f(x−t)+G(x+t)−G(x−t)## uses lower case f and capital G, those aren't like...

Homework Statement Show that the boundary-value problem $$u_{tt}=u_{xx}\qquad u(x,0)=2f(x)\qquad u_t(x,0)=2g(x)$$ has the solution $$u(x,t)=f(x+t)+f(x-t)+G(x+t)-G(x-t)$$ where ##G## is an antiderivative/indefinite integral of ##g##. Here, we assume that ##-\infty<x<\infty## and ##t\geq 0##...