[ 偏微分方程 2020 Spring ] 習題
Assignment 1 ( Done March 12, 2020 )
Problem 1
Derive the minimal surface equation
\[\left(1+u_y^2\right)u_{xx}-2u_xu_yu_{xy}+\left(1+u_x^2\right)u_{yy}=0\]
by the variational method.
Problem 2
Show that the problem
\[
\left\{\begin{array}{ll}
u_t=-u_{xx},& 0<x<\pi\\
u(0,t)=u(\pi,t)=0,& t\geq 0\\
u(x,0)=f(x), &0\leq x\leq \pi,
\end{array}
\right.
\]
is not well-posed.
Assignment 2 ( Done March 19, 2020 )
Problem 1
Solve the quasilinear equation
\[
\left \{
\begin{array}{l}
uu_{x}+u_{y}=1, \\
u(2s^{2},2s)=0,\quad s>0.
\end{array}
\right.
\]
Problem 2
Solve the quasilinear equation
\[
\left \{
\begin{array}{l}
(y^{2}-u^{2})u_{x}-xyu_{y}=xu,\quad x>0,\\
u(x\text{$,$}y=x)=x.
\end{array}
\right.
\]
Problem 3
\[
\left\{\begin{array}{ll}
u_{tt}-c^2u_{xx}=0,& 0<x<L, t>0\\
u(x,0)=f(x),& 0\leq x\leq L\\
u_t(x,0)=g(x), & 0\leq x\leq L
\end{array}
\right.
\]
and the boundary value problem
\[
\left\{\begin{array}{ll}
u_{tt}-c^2u_{xx}=0,& 0<x<L,\, t>0\\
u(0,t)=\alpha(t),& t\geq 0\\
u(L,t)=\beta(t), & t\geq 0,
\end{array}
\right.
\]
\left\{\begin{array}{ll}
u_{t}=u_{xx}+f(x,t), & \text{for }-\infty <x<\infty ,\text{ }t>0, \\
u(x,0)=\phi (x), & \text{for }-\infty <x<\infty , \\
u(x,t)\rightarrow 0, & \text{as }\left \vert x\right \vert \rightarrow \infty ,
\,\,t>0,
\end{array}
\right.
\]
\begin{array}{ll}
y^{\prime \prime }\left( x\right) +\lambda y\left( x\right) =0, & 0<x<1, \\
y\left( 0\right) =0, & \\
y\left( 1\right) +ky^{\prime }\left( 1\right) =0, & k>0\text{ is a constant.}
\end{array}
(a) Show that the above boundary value problem has infinitely
many positive eigenvalues
\[
\lambda _{0}<\lambda _{1}<\lambda _{2}<\cdots <\lambda _{n}<\cdots
\]
with \(\lim_{n\rightarrow \infty }\lambda _{n}=\infty \).
(b) Show that for each \(n\) the corresponding eigenfunction \(\phi_{n}\) has exactly \(n\) zeros in the interval \(\left( 0,1\right) \).
Problem 2
Consider the periodic Sturm-Liouville eigenvalue problem%
\[
\left \{
\begin{array}{l}
\left. y^{\prime \prime }+\lambda y=0,\;-\pi \leq x\leq \pi ,\right. \\
y(-\pi )=y(\pi ),\,y^{\prime }(-\pi )=y^{\prime }(\pi )\text{ (periodic end conditions).}
\end{array}
\right.
\]
(a) Show that the eigenvalues of the periodic Sturm-Liouville eigenvalue problem are \(0,\{n^{2}\}\), and the corresponding eigenfunctions are \(1,\) \(\{ \cos nx\},\) \(\{ \sin nx\}\), where \(n\) is a positive integer.
\begin{array}{cc}
\text{eigenvalue} & \text{eigenfunction(s)} \\
\lambda =0 & 1 \\
\lambda =n^{2} & \cos nx,\; \sin nx
\end{array}
(b) Compare it with Theorem 7.1.7.
Problem 3
Consider the regular Sturm-Liouville eigenvalue problem:
\[
\frac{d}{dx}\left( p\frac{dy}{dx}\right) +(q+\lambda s)y=0\text{,}
\]
\[
a_{1}y(a)+a_{2}y^{\prime }(a)=0,\; \ b_{1}y(b)+b_{2}y^{\prime }(b)=0
\]
in Section 7.1.2., where
(1) \(p\in C^{1}[a,b],\) \(q\in C[a,b],\) \(s\in C[a,b]\),
(2) \(p(x)>0\) and \(s(x)>0\) on \([a,b]\),
(3) \(a_{1}^{2}+a_{2}^{2}\neq 0\),
(4) \(b_{1}^{2}+b_{2}^{2}\neq 0\)
Problem 1
Derive the minimal surface equation
\[\left(1+u_y^2\right)u_{xx}-2u_xu_yu_{xy}+\left(1+u_x^2\right)u_{yy}=0\]
by the variational method.
Problem 2
Show that the problem
\[
\left\{\begin{array}{ll}
u_t=-u_{xx},& 0<x<\pi\\
u(0,t)=u(\pi,t)=0,& t\geq 0\\
u(x,0)=f(x), &0\leq x\leq \pi,
\end{array}
\right.
\]
is not well-posed.
Assignment 2 ( Done March 19, 2020 )
Problem 1
Solve the quasilinear equation
\[
\left \{
\begin{array}{l}
uu_{x}+u_{y}=1, \\
u(2s^{2},2s)=0,\quad s>0.
\end{array}
\right.
\]
Problem 2
Solve the quasilinear equation
\[
\left \{
\begin{array}{l}
(y^{2}-u^{2})u_{x}-xyu_{y}=xu,\quad x>0,\\
u(x\text{$,$}y=x)=x.
\end{array}
\right.
\]
Problem 3
Let \(u\) be a solution of
\[
a(x,y)u_{x}+b(x,y)u_{y}=-u
\]
of class \(C^{1}\) in the closed unit disk \(\Omega\) in the \(xy\)-plane. Let \(a(x,y)x+b(x,y)y\geq 0\) on the boundary of \(\Omega \). Prove that \(u\) vanishes identically.
Problem 4
\[
a(x,y)u_{x}+b(x,y)u_{y}=-u
\]
of class \(C^{1}\) in the closed unit disk \(\Omega\) in the \(xy\)-plane. Let \(a(x,y)x+b(x,y)y\geq 0\) on the boundary of \(\Omega \). Prove that \(u\) vanishes identically.
Problem 4
(a) Solve Euler's equation
\[
xu_{x}+yu_{y}+zu_{z}=\alpha u\quad(\alpha =\text{const.}\neq 0),
\]
\[u(x,y,z=1)=h(x,y).\]
\[
xu_{x}+yu_{y}+zu_{z}=\alpha u\quad(\alpha =\text{const.}\neq 0),
\]
\[u(x,y,z=1)=h(x,y).\]
Show that \(u\) is a homogeneous function of degree \(\alpha \); that is, \(u\) satisfies
\[
u(\lambda x,\lambda y,\lambda z)=\lambda ^{\alpha }u(x,y,z)
\]
for any \(\lambda >0\).
(b) Show that, for \(\alpha <0\), the only solution of (a) of class \(C^{1}\) in a neighborhood of the origin is the trivial solution \(u\equiv 0\).
Assignment 3 ( Done March 26, 2020 )
Problem 1
Check \[
w_{\nu \nu }+w_{\eta \eta }=-\frac{1}{3\eta }w_{\eta }.
\]
\[
4u_{xx}+5u_{xy}+u_{yy}+u_{x}+u_{y}=2.
\]
\[
u(\lambda x,\lambda y,\lambda z)=\lambda ^{\alpha }u(x,y,z)
\]
for any \(\lambda >0\).
(b) Show that, for \(\alpha <0\), the only solution of (a) of class \(C^{1}\) in a neighborhood of the origin is the trivial solution \(u\equiv 0\).
Assignment 3 ( Done March 26, 2020 )
Problem 1
Check \[
w_{\nu \nu }+w_{\eta \eta }=-\frac{1}{3\eta }w_{\eta }.
\]
Problem 2Determine the general solution of
\[
4u_{xx}+5u_{xy}+u_{yy}+u_{x}+u_{y}=2.
\]
Assignment 4 ( Done April 9, 2020 )
Problem 1
Show that there is at most one solution of the initial value problem\[
\left\{\begin{array}{ll}
u_{tt}-c^2u_{xx}=0,& 0<x<L, t>0\\
u(x,0)=f(x),& 0\leq x\leq L\\
u_t(x,0)=g(x), & 0\leq x\leq L
\end{array}
\right.
\]
and the boundary value problem
\[
\left\{\begin{array}{ll}
u_{tt}-c^2u_{xx}=0,& 0<x<L,\, t>0\\
u(0,t)=\alpha(t),& t\geq 0\\
u(L,t)=\beta(t), & t\geq 0,
\end{array}
\right.
\]
where \(c\) is a positive constant and we assume the compatibility conditions \(\alpha(0)=f(0)\) and \(\beta(0)=f(L)\).
\[u_{tt}-c^2u_{xx}=0.\]
Suppose \(\widetilde{u}\) is the solution of
\[
\left\{\begin{array}{ll}
u_{tt}-c^2u_{xx}=\widetilde{\varphi}(x,t),& x\in\mathbb{R},\, t\in\mathbb{R}^+\\
u(x,0)=\widetilde{f}(x),& x\in\mathbb{R}\\
u_t(x,0)=\widetilde{g}(x), & x\in\mathbb{R}
\end{array}
\right.
\]
where the following are satisfied
Problem 2
Let \(u\) be the solution of the wave equation\[u_{tt}-c^2u_{xx}=0.\]
Suppose \(\widetilde{u}\) is the solution of
\[
\left\{\begin{array}{ll}
u_{tt}-c^2u_{xx}=\widetilde{\varphi}(x,t),& x\in\mathbb{R},\, t\in\mathbb{R}^+\\
u(x,0)=\widetilde{f}(x),& x\in\mathbb{R}\\
u_t(x,0)=\widetilde{g}(x), & x\in\mathbb{R}
\end{array}
\right.
\]
where the following are satisfied
- \(\Vert f(\cdot)-\widetilde{f}(\cdot)\Vert_\infty <\epsilon\)
- \(\Vert g(\cdot)-\widetilde{g}(\cdot)\Vert_\infty <\epsilon\)
- \(\Vert \varphi(\cdot,t)-\widetilde{\varphi}(\cdot,t)\Vert_\infty <\epsilon\)
where \(\epsilon>0\) is a small constant. Prove that for fixed \(T>0\), \(u-\widetilde{u}\rightarrow 0\) uniformly in \(x\) on \([0,T]\).
\[u_{tt}-c^2u_{xx}=0,\quad 0<x<1,\,t>0\]
with initial conditions
\[
\left\{\begin{array}{ll}
u(x,0)=\sin \pi x,& 0<x<1\\
u_t(x,0)=0,& 0<x<1
\end{array}
\right.
\]
and boundary conditions
\[
\left\{\begin{array}{ll}
u(0,t)=0,& t>0\\
u(1,t)=0,& t>0.
\end{array}
\right.
\]
with spherical symmetry about the origin has the form
\[u=\frac{F(r+ct)+G(r-ct)}{r},\quad r=\vert x\vert\]
with suitable \(F, G\).
(b) Show that the solution with initial data of the form \(u=0\), \(u_t=g(r)\); (\(g\) is an even function of \(r\)) is given by
\[u(r,t)=\frac{1}{2cr}\int_{r-ct}^{r+ct}\rho g(\rho)\,d\rho.\]
\[
\mathcal{F}^{-1}\left(e^{-s^{2}t}\right)=\frac{1}{\sqrt{2t}}e^{-x^{2}/4t}.
\]
Problem 2
Problem 3
Derive the solution explicitly of the initial-boundary value problem\[u_{tt}-c^2u_{xx}=0,\quad 0<x<1,\,t>0\]
with initial conditions
\[
\left\{\begin{array}{ll}
u(x,0)=\sin \pi x,& 0<x<1\\
u_t(x,0)=0,& 0<x<1
\end{array}
\right.
\]
and boundary conditions
\[
\left\{\begin{array}{ll}
u(0,t)=0,& t>0\\
u(1,t)=0,& t>0.
\end{array}
\right.
\]
Problem 4
(a) Show that for \(n=3\) the general solution of \[\square u=u_{tt}-c^2\Delta u=0,\]with spherical symmetry about the origin has the form
\[u=\frac{F(r+ct)+G(r-ct)}{r},\quad r=\vert x\vert\]
with suitable \(F, G\).
(b) Show that the solution with initial data of the form \(u=0\), \(u_t=g(r)\); (\(g\) is an even function of \(r\)) is given by
\[u(r,t)=\frac{1}{2cr}\int_{r-ct}^{r+ct}\rho g(\rho)\,d\rho.\]
Assignment 5 ( Done April 23, 2020 )
Problem 1
Show that \[
\mathcal{F}^{-1}\left(e^{-s^{2}t}\right)=\frac{1}{\sqrt{2t}}e^{-x^{2}/4t}.
\]
(i) Derive
\begin{aligned}
u(x,t) &=\int_{\mathbb{R}^{n}}K(x,y,t)\phi (y)\,dy \\
&=\int_{\mathbb{R}^{n}}(4k\pi t)^{-n/2}e^{-\left \vert x-y\right \vert
^{2}/4kt}\phi (y)\,dy
\end{aligned}
by Fourier transform
\begin{aligned}
u(x,t) &=\int_{\mathbb{R}^{n}}K(x,y,t)\phi (y)\,dy \\
&=\int_{\mathbb{R}^{n}}(4k\pi t)^{-n/2}e^{-\left \vert x-y\right \vert
^{2}/4kt}\phi (y)\,dy
\end{aligned}
by Fourier transform
\[
\hat{f}(\xi )=\mathcal{F}(f)=(2\pi )^{-n/2}\int_{\mathbb{R}^{n}}e^{-ix\cdot\xi }f(x)\,dx.
\]
\hat{f}(\xi )=\mathcal{F}(f)=(2\pi )^{-n/2}\int_{\mathbb{R}^{n}}e^{-ix\cdot\xi }f(x)\,dx.
\]
(ii) Show that
- \[K(x,y,t)\in C^{\infty }\,\,\text{ for }\,\,x\in \mathbb{R}^{n},\text{ }y\in \mathbb{R}^{n},\text{ }t>0. \]
- \[\left( \frac{\partial }{\partial t}-k\Delta _{x}\right) K(x,y,t)=0\,\,\text{ for }\,\,t>0. \]
- \[K(x,y,t)>0\,\,\text{for}\,\,t>0. \]
- \[\int_{\mathbb{R}^{n}}K(x,y,t)\,dy=1\,\,\text{for }\,\,x\in \mathbb{R}^{n},\text{ }%t>0. \]
- For any \(\delta >0\) we have\[\lim \limits_{t\rightarrow 0^{+}}\int_{\left \vert y-x\right \vert >\delta }K(x,y,t)\,dy=0. \]
uniformly for \(x\in \mathbb{R}^{n}\). That is,\[\lim \limits_{t\rightarrow 0^{+}}\int_{\left \vert y-x\right \vert \leq \delta}K(x,y,t)\,dy=1.\]
Problem 3
Construct a solution of the nonhomogeneous 1-dimensional heat equation
\[\left\{\begin{array}{ll}
u_{t}=u_{xx}+f(x,t), & \text{for }-\infty <x<\infty ,\text{ }t>0, \\
u(x,0)=\phi (x), & \text{for }-\infty <x<\infty , \\
u(x,t)\rightarrow 0, & \text{as }\left \vert x\right \vert \rightarrow \infty ,
\,\,t>0,
\end{array}
\right.
\]
by Fourier transform.
Assignment 6 ( Done May 28, 2020 )
Problem 1
Let \(\Omega\) denote the unbounded set \(\left \vert x\right \vert >1\).
Assignment 6 ( Done May 28, 2020 )
Problem 1
Let \(\Omega\) denote the unbounded set \(\left \vert x\right \vert >1\).
(a) Let \(u\in C^{2}(\overline{\Omega })\), \(\Delta u=0\) in \(\Omega \) and \(\lim_{x\rightarrow \infty }u(x)=0\). Show that
\begin{aligned}
\max_{\overline{\Omega }}\left \vert u\right \vert =\max_{\partial \Omega
}\left \vert u\right \vert .
\end{aligned}
\begin{aligned}
\max_{\overline{\Omega }}\left \vert u\right \vert =\max_{\partial \Omega
}\left \vert u\right \vert .
\end{aligned}
(b) If the condition \(\lim_{x\rightarrow \infty }u(x)=0\) in part (a) is deleted, does the result still hold? Give a counterexample or a proof.
Problem 2
Show that when \(n=2\), \(x=(r\cos \theta ,r\sin\theta )\), \(y=(R\cos \phi ,R\sin \phi )\), then
\[
u(x)=\left \{
\begin{array}{ll}
\displaystyle\frac{R^{2}-\left \vert x\right \vert ^{2}}{\omega _{n}R}\int_{\partial B}\frac{\varphi (y)}{\left \vert x-y\right \vert ^{n}}\,ds_{y} & \text{for}\, x\in B \\
\varphi (x) & \text{for}\, x\in \partial B
\end{array}
\right.
\]
reduces to
\begin{aligned}
u(r,\theta )=\frac{1}{2\pi }\int_{0}^{2\pi }\frac{(R^{2}-r^{2})\varphi (\phi)}{R^{2}+r^{2}-2Rr\cos (\theta -\phi )}d\phi \,\text{for }r<R.
\end{aligned}
Problem 3
Derive
\begin{aligned}
\frac{\partial G}{\partial \nu }=\frac{\partial G}{\partial \left \vert
x\right \vert }=\frac{R^{2}-\left \vert y\right \vert ^{2}}{\omega _{n}R}
\left \vert x-y\right \vert ^{-n}\left \{
\begin{array}{l}
>0\,\,\text{if}\ y\in B_{R}\text{ }(y\neq x), \\
=0\,\,\text{ if}\ y\in \partial B_{R}\text{ }(y\neq x).
\end{array}
\right.
\end{aligned}
Problem 4
Recall that the Liouville theorem says that a harmonic function defined over \(\mathbb{R}^{n}\) \((n\geq 2)\) and bounded above is constant.
(a) Suppose \(u\in C(\bar{B}_{R})\) is a nonnegative harmonic function in \(B_{R}=B_{R}(x_{0})\). Show that
\[
\left \vert Du(x_{0})\right \vert \leq \frac{n}{R}u(x_{0}).
\]
(b) Prove the Liouville theorem by (a).
(c) Prove Liouville's theorem by the Harnack inequality
\[
\frac{R^{n-2}(R-\vert x\vert)}{(R+\vert x\vert)^{n-1}}u(0)\leq u(x)\leq \frac{R^{n-2}(R+\vert x\vert)}{(R-\vert x\vert)^{n-1}}u(0)
\]
when \(n=2\).
Problem 5
(a) If \(n=2\) prove that the Liouville theorem (Corollary 1) in Section 6.5 is valid for subharmonic functions.
(b) If \(n>2\) prove that a bounded subharmonic function defined over \(\mathbb{R}^{n}\) need not be constant.
Problem 6
Suppose \(u\) is a function defined on \(\mathbb{R}^{n}\) such that
Problem 7
(Rellich's Theorem) Let nontrivial function \(u\) satisfy \(u\in C^{2}(\mathbb{R}^{n})\), \(\Delta u=0\) in \(\mathbb{R}^{n}\) (\(n\geq 1\)). Show
that
\[
\int_{\mathbb{R}^{n}}u^{2}\,dx
\]
does not exist.
Problem 2
Show that when \(n=2\), \(x=(r\cos \theta ,r\sin\theta )\), \(y=(R\cos \phi ,R\sin \phi )\), then
\[
u(x)=\left \{
\begin{array}{ll}
\displaystyle\frac{R^{2}-\left \vert x\right \vert ^{2}}{\omega _{n}R}\int_{\partial B}\frac{\varphi (y)}{\left \vert x-y\right \vert ^{n}}\,ds_{y} & \text{for}\, x\in B \\
\varphi (x) & \text{for}\, x\in \partial B
\end{array}
\right.
\]
reduces to
\begin{aligned}
u(r,\theta )=\frac{1}{2\pi }\int_{0}^{2\pi }\frac{(R^{2}-r^{2})\varphi (\phi)}{R^{2}+r^{2}-2Rr\cos (\theta -\phi )}d\phi \,\text{for }r<R.
\end{aligned}
Problem 3
Derive
\begin{aligned}
\frac{\partial G}{\partial \nu }=\frac{\partial G}{\partial \left \vert
x\right \vert }=\frac{R^{2}-\left \vert y\right \vert ^{2}}{\omega _{n}R}
\left \vert x-y\right \vert ^{-n}\left \{
\begin{array}{l}
>0\,\,\text{if}\ y\in B_{R}\text{ }(y\neq x), \\
=0\,\,\text{ if}\ y\in \partial B_{R}\text{ }(y\neq x).
\end{array}
\right.
\end{aligned}
Problem 4
Recall that the Liouville theorem says that a harmonic function defined over \(\mathbb{R}^{n}\) \((n\geq 2)\) and bounded above is constant.
(a) Suppose \(u\in C(\bar{B}_{R})\) is a nonnegative harmonic function in \(B_{R}=B_{R}(x_{0})\). Show that
\[
\left \vert Du(x_{0})\right \vert \leq \frac{n}{R}u(x_{0}).
\]
(b) Prove the Liouville theorem by (a).
(c) Prove Liouville's theorem by the Harnack inequality
\[
\frac{R^{n-2}(R-\vert x\vert)}{(R+\vert x\vert)^{n-1}}u(0)\leq u(x)\leq \frac{R^{n-2}(R+\vert x\vert)}{(R-\vert x\vert)^{n-1}}u(0)
\]
when \(n=2\).
Problem 5
(a) If \(n=2\) prove that the Liouville theorem (Corollary 1) in Section 6.5 is valid for subharmonic functions.
(b) If \(n>2\) prove that a bounded subharmonic function defined over \(\mathbb{R}^{n}\) need not be constant.
Problem 6
Suppose \(u\) is a function defined on \(\mathbb{R}^{n}\) such that
- \(u\) is bounded and continuous on the half-space \(x_{n}\geq 0\) of \(\mathbb{R}^{n}\),
- \(u=0\) on \(x_{n}=0\),
- \(u\) is harmonic in \(x_{n}>0\).
Problem 7
(Rellich's Theorem) Let nontrivial function \(u\) satisfy \(u\in C^{2}(\mathbb{R}^{n})\), \(\Delta u=0\) in \(\mathbb{R}^{n}\) (\(n\geq 1\)). Show
that
\[
\int_{\mathbb{R}^{n}}u^{2}\,dx
\]
does not exist.
Assignment 7 ( Done June 4, 2020 )
Problem 1
Consider Problem 1
\begin{array}{ll}
y^{\prime \prime }\left( x\right) +\lambda y\left( x\right) =0, & 0<x<1, \\
y\left( 0\right) =0, & \\
y\left( 1\right) +ky^{\prime }\left( 1\right) =0, & k>0\text{ is a constant.}
\end{array}
(a) Show that the above boundary value problem has infinitely
many positive eigenvalues
\[
\lambda _{0}<\lambda _{1}<\lambda _{2}<\cdots <\lambda _{n}<\cdots
\]
with \(\lim_{n\rightarrow \infty }\lambda _{n}=\infty \).
(b) Show that for each \(n\) the corresponding eigenfunction \(\phi_{n}\) has exactly \(n\) zeros in the interval \(\left( 0,1\right) \).
Problem 2
Consider the periodic Sturm-Liouville eigenvalue problem%
\[
\left \{
\begin{array}{l}
\left. y^{\prime \prime }+\lambda y=0,\;-\pi \leq x\leq \pi ,\right. \\
y(-\pi )=y(\pi ),\,y^{\prime }(-\pi )=y^{\prime }(\pi )\text{ (periodic end conditions).}
\end{array}
\right.
\]
(a) Show that the eigenvalues of the periodic Sturm-Liouville eigenvalue problem are \(0,\{n^{2}\}\), and the corresponding eigenfunctions are \(1,\) \(\{ \cos nx\},\) \(\{ \sin nx\}\), where \(n\) is a positive integer.
\begin{array}{cc}
\text{eigenvalue} & \text{eigenfunction(s)} \\
\lambda =0 & 1 \\
\lambda =n^{2} & \cos nx,\; \sin nx
\end{array}
(b) Compare it with Theorem 7.1.7.
Problem 3
Consider the regular Sturm-Liouville eigenvalue problem:
\[
\frac{d}{dx}\left( p\frac{dy}{dx}\right) +(q+\lambda s)y=0\text{,}
\]
\[
a_{1}y(a)+a_{2}y^{\prime }(a)=0,\; \ b_{1}y(b)+b_{2}y^{\prime }(b)=0
\]
in Section 7.1.2., where
(1) \(p\in C^{1}[a,b],\) \(q\in C[a,b],\) \(s\in C[a,b]\),
(2) \(p(x)>0\) and \(s(x)>0\) on \([a,b]\),
(3) \(a_{1}^{2}+a_{2}^{2}\neq 0\),
(4) \(b_{1}^{2}+b_{2}^{2}\neq 0\)
(a) Suppose that \(a_{1}a_{2}\leq 0,\) \(b_{1}b_{2}\geq 0,\) and \(q(x)\leq 0\). Show that \(\lambda =0\) is an eigenvalue implies \(a_{1}=b_{1}=0\).
(b) Suppose that \(a_{1}a_{2}\leq 0,\) \(b_{1}b_{2}\geq 0,\) and \(q(x)\equiv 0.\) Show that \(a_{1}=b_{1}=0\) implies \(\lambda =0\) is an eigenvalue.
Problem 4
Show that any second-order equation
\[
a_{0}(x)y^{\prime \prime }+a_{1}(x)y^{\prime }+a_{2}(x)y=0,\;(a_{0}\neq 0)
\]
can be transformed into normal form
\[
u^{\prime \prime }+q(x)u=0
\]
by the change of variables
\[
y=u\exp \left( -\frac{1}{2}\int^{x}\frac{a_{1}}{a_{0}}dt\right) ,
\]
where
\[
q(x)=\frac{a_{2}}{a_{0}}-\frac{1}{4}\frac{a_{1}^{2}}{a_{0}^{2}}-\frac{1}{2}
\frac{a_{1}^{\prime }a_{0}-a_{1}a_{0}^{\prime }}{a_{0}^{2}}.
\]
Problem 5
Suppose that \(q\in C[x_{1},x_{2}]\) and \(q\leq 0\) in \((x_{1},x_{2}).\) Show that any nontrivial solution \(\phi (x)\) of \(y^{\prime \prime }+q(x)y=0\) has at most one zero in \((x_{1},x_{2})\).
Problem 6
Show that Airy's equation \(y^{\prime \prime}(x)+xy(x)=0\) is oscillatory.
(b) Suppose that \(a_{1}a_{2}\leq 0,\) \(b_{1}b_{2}\geq 0,\) and \(q(x)\equiv 0.\) Show that \(a_{1}=b_{1}=0\) implies \(\lambda =0\) is an eigenvalue.
Problem 4
Show that any second-order equation
\[
a_{0}(x)y^{\prime \prime }+a_{1}(x)y^{\prime }+a_{2}(x)y=0,\;(a_{0}\neq 0)
\]
can be transformed into normal form
\[
u^{\prime \prime }+q(x)u=0
\]
by the change of variables
\[
y=u\exp \left( -\frac{1}{2}\int^{x}\frac{a_{1}}{a_{0}}dt\right) ,
\]
where
\[
q(x)=\frac{a_{2}}{a_{0}}-\frac{1}{4}\frac{a_{1}^{2}}{a_{0}^{2}}-\frac{1}{2}
\frac{a_{1}^{\prime }a_{0}-a_{1}a_{0}^{\prime }}{a_{0}^{2}}.
\]
Problem 5
Suppose that \(q\in C[x_{1},x_{2}]\) and \(q\leq 0\) in \((x_{1},x_{2}).\) Show that any nontrivial solution \(\phi (x)\) of \(y^{\prime \prime }+q(x)y=0\) has at most one zero in \((x_{1},x_{2})\).
Problem 6
Show that Airy's equation \(y^{\prime \prime}(x)+xy(x)=0\) is oscillatory.
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