1. 1. A linear differential operator $ℒ$ has the property that $ℒe^{m x} ≡ F(m) e^{m x}$, where $F(α)=F'(α)=0$ for some $α$. Show that the homogeneous differential equation $ℒ y=0$ is satisfied both by $y(x)=e^{α x}$ and by $y(x)=x e^{α x}$.
      Consider the boundary-value problem $$\tag{#}\begin{aligned} ℒ y(x)&=y''(x)+6 y'(x)+9 y(x)=f(x)\\ y(0)&=0\\ y(1)&=0 \end{aligned}$$
   2. In terms of the delta function, state the equation and boundary conditions satisfied by the Green's function $g(x, ξ)$ for the problem (#). Hence solve for $g$ and give the solution to (#) in terms of $g$.
   3. Find the eigenvalues $λ_n$ and eigenfunctions $y_n$ for the problem $ℒ y=λ y$, with the same boundary conditions as in (#). Find the adjoint operator $ℒ^*$ and adjoint boundary conditions, and thus find the eigenfunctions $w_n$ for the adjoint problem $ℒ^* w=λ w$. Thus obtain the solution to (#) as an eigenfunction expansion of the form $y(x)=\sum_i c_i y_i(x)$
      [Note that $λ_n<0$ for all n.]
   4. Show that the problem $$ ℒ y(x)+π^2 y(x)=f(x),   y(0)=1,   y(1)=2 $$ has no solution for $y$ unless $f$ satisfies a solvability condition, which you should state explicitly.
2. Consider the differential equation $$\tag† \left(1-x^2\right) y''(x)-x y'(x)+λ^2 y(x)=0 $$ where $λ$ is a real non-negative parameter.
   1. Classify $x= ± 1$ and $x=∞$ as ordinary, regular singular or irregular singular points of (†). Find the indicial equations, where appropriate, and thus give the leadingorder behaviour of two linearly independent solutions to (†) as $x → ± 1$ and as $x → ∞$, assuming that $2 λ$ is not an integer.
      Explain briefly what might go wrong if $2 λ$ is an integer.
   2. Seek the solution to (†) as a power series expansion about $x=0$, i.e. $$ y(x)=\sum_{k=0}^{∞} a_k x^k $$ Find a recurrence relation for the coefficients $a_k$ and thus show that (†) admits a polynomial solution $P_n(x)$ if and only if $λ$ is equal to an integer $n$.
   3. By transforming (t) into self-adjoint form, or otherwise, show that the polynomials $P_{\mathrm{n}}(x)$ satisfy the orthogonality conditions $$ ∫_a^b \frac{P_n(x) P_m(x)}{\sqrt{1-x^2}} \mathrm{~d} x=0   \text { whenever } m ≠ n, $$ where the limits $\{a, b\}$ of the domain are to be determined.
3. 1. When $0<ϵ ⩽ 1$ and $x>0$, explain why the nonlinear algebraic equation $$\tag⋆ \frac{ϵ y^2}{1+x y^2}+x^2 y=x^2 $$ has a unique positive solution $y(x ; ϵ)$. [Hint: consider how the left-hand side behaves as a function of $y$.]
      Write this positive solution as an asymptotic expansion of the form $$ y(x ; ϵ) ∼ y_0(x)+ϵ y_1(x)+⋯ $$ Calculate the first two terms $y_0$ and $y_1$, and deduce that the asymptotic expansion fails when $x=O\left(ϵ^{1 / 2}\right)$.
      By using the re-scaling, $x=ϵ^{1 / 2} X$ and $y(x)=Y(X)$, find the leading-order inner solution $Y_0(X)$, and demonstrate that it matches asymptotically with the leading-order outer solution $y_0(x)$.
   2. With $0<ϵ ≪ 1$, seek a solution to the nonlinear boundary value problem $$ y''(x)+\left(1+ϵ λ+ϵ y'(x)^2-ϵ y(x)^2\right) y(x)=0,   y(0)=0,   y(π)=0 $$ as an asymptotic expansion of the form $y(x) ∼ y_0(x)+ϵ y_1(x)+⋯$. Show that a possible leading-order solution is $y_0(x)=A \sin x$, where $A$ satisfies the equation $$ A\left(A^2-2 λ\right)=0 $$ [Hint: you may use without proof the identity $\sin ^3 x=\frac{3 \sin x-\sin (3 x)}{4}$.]
   3. Consider the boundary-value problem $$ ϵ y''(x)-\left(1+x^2\right) y'(x)+x y(x)=0,   y(0)=1,   y(1)=0 $$ where $ϵ$ is a small positive parameter. Find the leading-order outer solution $y_0(x)$ in an asymptotic expansion of the form $y(x) ∼ y_0(x)+ϵ y_1(x)+⋯$. Verify that it is impossible to satisfy both boundary conditions. Identify where there is a boundary layer in the solution, solve for the leading-order inner solution in the boundary layer, and asymptotically match the inner and outer solutions. Hence show that the gradient of the solution at $x=1$ is given approximately by $$ y'(1) ∼-\frac{2^{3 / 2}}{ϵ} . $$

Solution

1. 1. $ℒe^{m x} ≡ F(m) e^{m x}$, so $ℒe^{α x} = F(α) e^{α x}=0$ and $\frac∂{∂m}[ℒe^{mx}]=ℒ[\frac∂{∂m}e^{mx}]=ℒ[xe^{mx}]$
      $∴ℒ[xe^{mx}]=\left(F'(m)+xF(m)\right)e^{mx}$
      $∴ℒ[xe^{αx}]=(F'(α)+xF(α))e^{αx}=0$
   2. Green’s function satisfies $g_{xx}+6g_x+9g=δ(x-ξ)$ with $g=0$ at $x=0,1$.
      So we have $g_{xx}+6g_x+9g=0$ for $x≠ξ$
      Auxiliary equation $(m+3)^2=0$, so general solution $g=C_1e^{-3x}+C_2xe^{-3x}$
      Given boundary conditions$\begin{cases} A(ξ)xe^{-3x}&0<x<ξ<1\\ B(ξ)(1-x)e^{-3x}&0<ξ<x<1 \end{cases}$
      At $x=ξ$, jump conditions$\begin{cases} [g]_{ξ-}^{ξ+}=0\\ [g_x]_{ξ-}^{ξ+}=1 \end{cases}$