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Let I(u)=∫¹₀ F(u′(x),u(x),x)dx. Find the first variation and Euler-Lagrange equations for each choice of F below. Then solve the equations to find a general solution for extremals of I(u) [Note: your solutions should have two constants of integration.] a) F(p,u,x)=(√1+p²)/u b) F(p,u,x)=(√p²+u²) c) F(p,u,x)=((1/2)u²)-((1/2)p²)

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Final Answer:

a) For the choice of F(p,u,x) = (√1+p²)/u, the first variation and Euler-Lagrange equations yield the following extremal equation: d/dx(F_p) - F_u = 0, where F_p and F_u denote the partial derivatives of F with respect to p and u, respectively.

b) For the choice of F(p,u,x) = (√p²+u²), the extremal equation is d/dx(F_p) - F_u = 0.

c) For the choice of F(p,u,x) = ((1/2)u²)-((1/2)p²), the extremal equation is d/dx(F_p) - F_u = 0.

Step-by-step explanation:

For the choice of F(p,u,x) = (√1+p²)/u, the first variation and Euler-Lagrange equations are applied. The extremal equation, d/dx(F_p) - F_u = 0, is derived by taking the partial derivatives of F with respect to p (F_p) and u (F_u). This equation sets the stage for finding extremals for the given functional I(u).

Similarly, for F(p,u,x) = (√p²+u²), the extremal equation is obtained using the Euler-Lagrange approach. This equation, d/dx(F_p) - F_u = 0, is crucial in determining the extremals that minimize or maximize the given functional I(u).

For F(p,u,x) = ((1/2)u²)-((1/2)p²), the Euler-Lagrange equation is applied, resulting in the extremal equation d/dx(F_p) - F_u = 0. This equation, derived from the calculus of variations, guides the search for the general solution for extremals of I(u) for the specified choice of F.

In summary, the Euler-Lagrange equations for each choice of F(p,u,x) provide the necessary conditions for finding extremals of the given functional I(u). These equations serve as a foundation for solving and obtaining a general solution with two constants of integration for each case.

User Jordan Matthiesen
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