Problem 3
Determine second-order uniformly valid expansions for the problems $$ \begin{gathered} e y^{\prime \prime} \pm(2 x+1) y^{\prime}+2 y=0 \\ y(0)=\alpha, \quad y(1)=\beta \end{gathered} $$ using (a) the MMAE (b) Latta's method, and (c) the Bromberg-Visik-Lyusternik method.
Problem 5
Consider the problem $$ \begin{array}{r} e y^{\prime \prime}-y^{\prime}+y=0 \\ y(0)=\alpha, \quad y(1)=\beta, \quad y^{\prime}(1)=y \end{array} $$ (a) Show that boundary layers exist at both ends which are characterized by the stretching transformations $$ \eta=x / \epsilon \text { and } \zeta=(1-x) / \epsilon $$ (b) Determine a second-order uniformly valid solution using the MMAE. (c) Determine a second-order expansion using the MCE by letting $$ y=F(x ; \epsilon)+G(\eta ; \epsilon)+H(\zeta ; \epsilon) $$ where \(G \rightarrow 0\) as \(\eta \rightarrow \infty\) and \(H \rightarrow 0\) as \(\zeta \rightarrow \infty\)
Problem 7
Show that the MMAE cannot be used to obtain uniformly valid expansions for $$ \begin{gathered} \epsilon^{2} y^{\prime \prime}+y=f(x) \\ y(0)=\alpha, \quad y(1)=\beta \end{gathered} $$ Can you conclude from this example that the MMAE is inapplicable to oscillation problems?
Problem 9
The vibration of a beam with clamped ends is governed by $$ \begin{gathered} \epsilon^{2} \frac{d^{4} u}{d x^{4}}-\frac{d^{2} u}{d x^{2}}=\lambda^{2} u \\ u(0)=u(1)=u^{\prime}(0)=u^{\prime}(1)=0 \end{gathered} $$ Determine a first-order expansion for small \(\epsilon\) for \(u\) and \(\lambda\).
