4.4: Problems
- Page ID
- 106225
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4.1. Solve the following problem:
\[x^{\prime \prime}+x=2, \quad x(0)=0, \quad x^{\prime}(1)=0. \nonumber \]
4.2. Find product solutions, \(u(x, t)=b(t) \phi(x)\), to the heat equation satisfying the boundary conditions \(u_{x}(0, t)=0\) and \(u(L, t)=0\). Use these solutions to find a general solution of the heat equation satisfying these boundary conditions.
4.3. Consider the following boundary value problems. Determine the eigenvalues, \(\lambda\), and eigenfunctions, \(y(x)\) for each problem.
a. \(y^{\prime \prime}+\lambda y=0, \quad y(0)=0, \quad y^{\prime}(1)=0\).
b. \(y^{\prime \prime}-\lambda y=0, \quad y(-\pi)=0, \quad y^{\prime}(\pi)=0\).
c. \(x^{2} y^{\prime \prime}+x y^{\prime}+\lambda y=0, \quad y(1)=0, \quad y(2)=0\).
d. \(\left(x^{2} y^{\prime}\right)^{\prime}+\lambda y=0, \quad y(1)=0, \quad y^{\prime}(e)=0\).
4.4. For the following sets of functions: i) show that each is orthogonal on the given interval, and ii) determine the corresponding orthonormal set.
a. \(\{\sin 2 n x\}, \quad n=1,2,3, \ldots, \quad 0 \leq x \leq \pi\).
b. \(\{\cos n \pi x\}, \quad n=0,1,2, \ldots, \quad 0 \leq x \leq 2\).
c. \(\left\{\sin \dfrac{n \pi x}{L}\right\}, \quad n=1,2,3, \ldots, \quad x \in[-L, L]\).
4.5. Consider the boundary value problem for the deflection of a horizontal beam fixed at one end,
\[\dfrac{d^{4} y}{d x^{4}}=C, \quad y(0)=0, \quad y^{\prime}(0)=0, \quad y^{\prime \prime}(L)=0, \quad y^{\prime \prime \prime}(L)=0 \nonumber \]
Solve this problem assuming that \(C\) is a constant.