Partial Differential Equations: Exercises

17 Sturm-Liouville problems, part 2

Exercise 1: Identifying regular Sturm-Liouville problems

For each of the boundary value problems below, identify which are regular Sturm-Liouville problems, which are singular Sturm-Liouville problems, and which are neither. For those that are Sturm-Liouville problems, identify all the relevant components.

  1. \(xf''(x) + f'(x) + \left( -\frac{1}{x} + \lambda x\right) f(x) = 0\) on the interval \([0, 1]\) with the boundary conditions \(f(0)=0\) and \(f(1)=0\).

  2. \((1-x^2)f''(x) -2x f'(x) + \lambda f(x)=0\) on the interval \([-1,1]\) with boundary conditions \(f(-1)=f(1)\) and \(f'(-1)=f'(1)\).

  3. \(x^3 f''(x) + 3x^2 f'(x) + \lambda f'(x) f(x) = 0\) on the interval \([0, 2]\) with boundary conditions \(f(0)=f(2)\) and \(f'(0)=f'(2)\).

  4. \((xf'(x))' + \frac{\lambda}{x}f(x) =0\) on the interval \([1,2]\) with boundary conditions \(f(1)=0\) and \(f(2)=0\).

  1. This is a singular Sturm-Liouville problem. We have \(r(x)=x\), \(p(x) = -1/x\), and \(w(x)=x\) with separated boundary conditions. But notice that \(r(0)=0\) and \(w(0)=0\), as well as that \(p(0)\) is undefined, which cannot happen for regular problems.

  2. This is another singular Sturm-Liouville problem. We have \(r(x) = 1-x^2\), \(p(x) = 0\), and \(w(x)=1\) with periodic boundary conditions. But notice that \(r(\pm 1)\), which cannot happen for regular problems.

  3. This is neither a regular nor a singular Sturm-Liouville problem. In fact, notice that it is not even a linear differential equation because of the product \(f'(x)f(x)\).

  4. This is a regular Sturm-Liouville problem. We have \(r(x) = x\), \(p(x)=0\), and \(w(x) = 1/x\). The boundary conditions are separated.