Written Qualifying Examination in DIFFERENTIAL EQUATIONS September 14, 1995

Suppose \(D\) is the rectangle defined by \(\lvert x-x_0\rvert \le a\), \(\lvert y-y_0\rvert \le b\) in the plane \(\mathbb {R}^2\). Suppose that \(f\) is a real valued function defined on \(D\) which is continuous, hence bounded by some nonnegative number \(M\) on \(D\).

1. Show that if \(\alpha =\min \{a,b/M\}\), then there exists a function \(y(x)\), defined on \(\lvert x-x_0\rvert \le \alpha \), such that \[ y'(x)=f(x,y(x))\quad \text {and}\quad y(x_0)=y_0. \]
2. Show that the solutions \(y\) guaranteed in part (i) may not be unique; that is, give an example in which (i) holds and more than one solution \(y(x)\) exists through the point \((x_0,y_0)\) in your example.

1. Suppose \(A\) is a constant \(n\times n\) matrix and consider the equation \[ \dot {x}(t)=Ax(t). \] Show that if all eigenvalues have negative real parts then the trivial solution \(x(t)\equiv 0\) is asymptotically stable; that is, every solution \(x(t)\to 0\) as \(t\to \infty \).
2. Show that if one eigenvalue of \(A\) has positive real part, then the trivial solution of \(A\) is not stable.
3. Under what circumstances is the trivial solution stable if \(A\) has no eigenvalue with positive real part, but does have eigenvalues with zero real part? (State, don’t prove)
4. Do any of these results carry over if \(A\) is not a constant matrix? (State, don’t prove.)

(Concrete cases from 2.) Change to vector-matrix form and discuss the stability of the trivial solution of the following equations.

1. \(u''-u=0\)
2. \(u''+2ku'+u=0\) with \(k>0\)
3. \(u^{(4)}+2u''+u=0\)
4. \(y'=\Theta y\) where \(\Theta \) is the matrix of all zeros.

1. Suppose \(A\) is an \(n\times n\) matrix whose elements \(a_{ij}(t)\) are each of period \(T\). You may assume existence and uniqueness of a fundamental matrix \(\Phi (t)\) for the equation \[ \dot {x}(t)=A(t)x(t). \] Show that \(\Phi (t+T)\) is also a fundamental matrix for the equation, hence \[ \Phi (t+T)=\Phi (t)\cdot C \] for some non-singular matrix \(C\).

2. You may assume that a non-singular matrix \(C\) has a logarithm, i.e., \(C=e^B\) for some \(n\times n\) matrix \(B\).

   Show that there exists a non-singular matrix \(P(t)\) of period \(T\) such that \[ \Phi (t)=P(t)e^{Bt} \qquad \text {Floquet's Theorem.} \]

   Hint: Solve for \(P(t)\) and check.

3. Make the change of variables \(y=P(t)u(t)\) and change the periodic system \(y'=A(t)y\) into the system \[ u'=Bu. \]

Write a short essay discussing the basic concepts of autonomous dynamical systems. Describe flows, trajectories, equilibrium points, stability, asymptotic stability and local linearizations, etc.

Write a brief discussion of the method of characteristics and then demonstrate the method by solving \[ u_x+u_y=u^2;\qquad u(x,0)=x^2. \]

Suppose \(u(x,t): x\in \mathbb {R}\), \(t\in [0,\infty )\) is of class \(C^\infty \) and that \(u\) is a solution of \[ \begin {cases} u_{tt}=u_{xx}\\ u(x,0)=g(x)\\ u_t(x,0)=h(x). \end {cases} \] Suppose \(g\) and \(h\) are in \(C_0^\infty \). (\(C^\infty \) functions with compact support.) Define \[ k(t)=\frac 12\int _{-\infty }^{\infty }u_t^2(x,t)\,dx\quad \text {(kinetic energy)} \] and \[ p(t)=\frac 12\int _{-\infty }^{\infty }u_x^2(x,t)\,dx\quad \text {(potential energy)} \] Prove that \(k(t)+p(t)\) is constant and that for sufficiently large times \(t\), \(k(t)=p(t)\). Hint: d’Alembert’s solution.

Consider the Sturm-Liouville operator \[ \mathcal {L} u=\frac {1}{w(x)}\{-(p(x)u')'+q(x)u\} \] where \(p(x)>0\), \(w(x)>0\) and \(1/p(x)\), \(q(x)\) and \(w(x)\) are locally Lebesgue integrable on an arbitrary interval \(I\) (open or closed, bounded or unbounded).

1. Define what is meant by a solution of \(\mathcal {L} u=\lambda u\) on \(I\).
2. Show that the initial value problem for the differential equation \(\mathcal {L} u=\lambda u\) on \(I\) with \(u(x_0)=c_1\), \((pu')(x_0)=c_2\) at \(x_0\in I\) is equivalent to an integral equation for the vector \((u_1,u_2)=(u,pu')\). Prove the equivalence.

A bounded linear operator \(P\) in a complex Hilbert space \(\mathcal {H}\) is a projection if and only if \(P^2=P\). Prove that a projection \(P\) is an orthogonal projection (so that \(P\mathcal {H}\perp (1-P)\mathcal {H}\)) if and only if \(P\) is selfadjoint. (You may use the known theorem that a bounded linear operator \(A\) in \(\mathcal {H}\) is selfadjoint if and only if \((Ax,x)\) is real for all \(x\in \mathcal {H}\).)

Define a regular Sturm-Liouville operator by the equation \[ \mathcal {L} u=-(2x)^{-1}\bigl ((2x)^{-1}u'\bigr )' \] on \(1\le x\le \sqrt {2}\) using boundary conditions \(u(1)=u(\sqrt {2})=0\).

1. Explain how \(\mathcal {L}\) and the boundary conditions can be used to construct a selfadjoint operator \(S\) in a suitable Hilbert space \(\mathcal {H}\). Do not give proofs.
2. Calculate the eigenvalues and eigenfunctions of \(S\) using the solution basis \(u_1(x)=\sin (\sqrt {\lambda }x^2)\), \(u_2(x)=\cos (\sqrt {\lambda }x^2)\) of the differential equation \(\mathcal {L} u-\lambda u=0\). Do not derive the basis.
3. Verify by direct integration that the eigenfunctions are orthogonal in \(\mathcal {H}\).

Let \(S\) be the selfadjoint realization in the Hilbert space \(\mathcal {H}=L_2(0,\infty )\) for the singular Sturm-Liouville operator \(\mathcal {L}=-u''\) on \(0<x<\infty \) with boundary condition \(u'(0)=0\). Assume that \(S\) is defined by the domain \[ D(S)=\{u\in \mathcal {H}:u''\in \mathcal {H}\text { and }u'(0)=0\} \] and the relation \(Su=-u''\) for all \(u\in D(S)\).

1. Show that the Green’s function \(G\) for the operator \(S\) is given by the equation below, where \(\zeta =re^{i\theta }\) and \(\sqrt {\zeta }=\sqrt {r}e^{i\theta /2}\) on \(0<\theta <2\pi \): \[ G(x,y,\zeta )= \begin {cases} i e^{i\sqrt {\zeta }y}\cos (\sqrt {\zeta }x)/\sqrt {\zeta }, & 0\le x\le y<\infty ,\\ i e^{i\sqrt {\zeta }x}\cos (\sqrt {\zeta }y)/\sqrt {\zeta }, & 0\le y\le x<\infty . \end {cases} \]
2. Prove that the operator \(S\) has a pure continuous spectrum consisting of all non-negative real numbers.