Question

Solve the extremes in the following questions using Euler's condition. J=∫F(x(t),x ′ (t))dt is the objective function. Stated as dx/dt=x ′ ,x=x(t) ′

a. F=(x)2−k2×2 ( k is fixed number )

Answer 1

Euler's condition is applied to the provided functional form to derive Euler's differential equation, which, when solved, yields the extremal solution for the variational problem.

Step-by-step explanation:

The subject question relates to solving an extremal problem using Euler's condition. The objective function given is J=∫F(x(t),x'(t))dt, with the specific functional form F=(x)^2−k^2×(x′)^2. According to Euler's equation for calculus of variations, the necessary condition for an extremal is that the first variation of the integral is zero. This leads to Euler's differential equation:

∂F/∂x − d(∂F/∂x')/dt = 0.

Applying this to the function provided, one would first compute the partial derivatives of F with respect to x and x', and then solve the ordinary differential equation (ODE) that emerges from Euler's condition. The fixed number k acts as a parameter. Without additional boundary conditions, solving for explicit solutions of x(t) may involve integration of the resulting ODE, and the use of constants of integration to accommodate potential initial or boundary values.

Answer 2

To solve for extremals using Euler's condition for the integral J with F=(x)^2−k^2×^2, we derive the Euler-Lagrange equation and solve the resulting second-order differential equation.

Step-by-step explanation:

To solve the extremals of the given integral J=∫F(x(t),x' (t))dt where F=(x)^2−k^2×^2 and k is a fixed number, we apply Euler's condition from the calculus of variations. Euler's condition gives us the necessary condition for x(t) to be an extremal of J.

According to Euler's condition, for a functional of the form J=∫F(x,x')dt, the extremal function x(t) must satisfy the Euler-Lagrange equation d/dt(∂F/∂x') - ∂F/∂x = 0. Applying this to our functional with F=(x)^2−k^2×^2, we get:

∂F/∂x = 2x

∂F/∂x' = -2k^2x'

d/dt(∂F/∂x') = -2k^2x''

Substituting these into the Euler-Lagrange equation, we have -2k^2x'' - 2x = 0. This is a second-order linear homogeneous differential equation. To solve it, we find the characteristic equation and its roots. Assuming a solution of the form x(t) = e^λt, substituting into the differential equation gives us a quadratic equation in λ.

From here, the solution proceeds by finding the general solution to the differential equation, which involves constants that would be determined by boundary conditions.