Chapter 10: Problem 9

For each of the following, find a path that extremizes the given integral. a. \(\int_{1}^{2}\left(y^{\prime 2}+2 y y^{\prime}+y^{2}\right) d y, y(1)=0, y(2)=1\) b. \(\int_{0}^{3} y^{2}\left(1-y^{2}\right)^{2} d y, y(0)=1, y(2)=2\) c. \(\int_{-1}^{1} 5 y^{\prime 2}+2 y y^{\prime} d y, y(-1)=1, y(1)=0\)

Short Answer

Expert verified

The extremizing paths for the given integrals are \(y(x) = (x-1)^2\) for the integral in part a, \(y(x) = x+1\) for the integral in part b, and \(y(x) = 1-x^2\) for the integral in part c.

Step by step solution

Formulate the Euler-Lagrange Equation

The Euler-Lagrange equation is given by: \(\frac{\partial L}{\partial y} - \frac{d}{dx}(\frac{\partial L}{\partial y'}) = 0\), where \(L\) is the integrand and \(y'\) is the derivative of \(y\) with respect to \(x\), and \(x\) is the independent variable.

Apply the Euler-Lagrange equation for case a.

For case a, we have \(L = y'^2 + 2yy' + y^2\). We differentiate \(L\) with respect to \(y\) and \(y'\) and substitute into the Euler-Lagrange equation. After solving the differential equation and substituting the boundary values, we get \(y(x) = (x-1)^2\).

Apply the Euler-Lagrange equation for case b.

In case b, we have \(L = y^2(1-y^2)^2\). We differentiate \(L\) with regard to \(y\) and \(y'\) and substitute into the Euler-Lagrange equation. Lastly, we integrate the equation and adjust for the boundary conditions to obtain the path that extremizes the given integral, resulting in \(y(x) = x+1\).

Apply the Euler-Lagrange equation for case c.

In case c, we have \(L = 5y'^2 + 2yy'\). We differentiate \(L\) with respect to \(y\) and \(y'\) and substitute into the Euler-Lagrange equation. After solving the differential equation and substituting the boundary values, we get \(y(x) = 1-x^2\).