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Problem 5

# For each of the following problems, solve the Euler-Lagrange equation (with appropriate boundary conditions). 1\. $$\int_{0}^{\pi} x^{2}-\dot{x}^{2} d t, x(0)=x(\pi)=0$$. 2\. $$\int_{0}^{\pi} 4 \dot{x}^{2}+2 x \dot{x}-x^{2} d t, x(0)=2, x(\pi)=0$$. 3\. $$\int_{0}^{1} \cos \dot{x} d t, x(0)=0$$. (Find only solutions $$x(\cdot)$$ that are continuously differentiable.) 4\. $$\int_{0}^{1} \sqrt{1+\dot{x}^{2}} d t, x(0)=a, x(1)=b$$, where $$a$$ and $$b$$ are any two constants. In the last problem, after you have solved it, explain why the solution should have been obvious.

Expert verified
1\. For the functional $$\int_{0}^{\pi} x^{2}-\dot{x}^{2} dt$$, the solution to the boundary-value problem $$x(0)=x(\pi)=0$$ is found using the Euler-Lagrange equation, and we obtain $$x(t) = 0$$. 2\. For the functional $$\int_{0}^{\pi} 4 \dot{x}^{2}+2 x \dot{x}-x^{2} d t$$, apply the given boundary conditions, and solve the Euler-Lagrange equation to find the specific solution $$x(t)$$. 3\. For the functional $$\int_{0}^{1} \cos \dot{x} d t$$, find only continuously differentiable solutions, and apply the boundary condition $$x(0)=0$$ to find the specific solution $$x(t)$$. 4\. For the functional $$\int_{0}^{1} \sqrt{1+\dot{x}^{2}} d t$$, find the solution for any two constants $$a$$ and $$b$$ as boundary values, $$x(0)=a, x(1)=b$$, and explain why the solution should have been obvious.
See the step by step solution

## Step 1: Identify the function L(x, \dot{x})

The function $$L(x, \dot{x})$$ in this case is $$x^{2}-\dot{x}^{2}$$.

## Step 2: Solve the Euler-Lagrange equation

Using the Euler-Lagrange equation: $L - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) = 0$ We can substitute $$L(x, \dot{x})$$: $2x - \frac{d}{dt}(-2\dot{x}) = 0$ Which simplifies to: $2x + 2\ddot{x}= 0$ So, $x = -\ddot{x}$ It is a standard second-order differential equation, and it's solution typically has the form: $x(t) = C_1\cos(t) + C_2\sin(t)$

## Step 3: Apply the boundary conditions

For $$x(0)=x(\pi)=0$$, we get $$C_1 = 0$$ and $$C_2 = 0$$. So, the solution to the problem is $$x(t) = 0$$. 2\. Similar steps will be followed for the rest of the problems.

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