Test Bank

forOrdinary Differential Equations

June, 2001

Part I: General Theory.

(#1) (Gronwall’s inequality).(a) Suppose α and β are nonnegative constants, and φ(·) is a continuousfunction defined on an interval [a, b] satisfying

φ(t) ≤ α + β∫ t

aφ(s) ds

for all t ∈ [a, b]. State and prove Gronwall’s inequality.(b) Suppose α(·) and β(·) are now nonnegative continuous functions definedon [a, b], and φ(·) satisfies

φ(t) ≤ α(t) +∫ t

aβ(s)φ(s) ds.

Show that

φ(t) ≤ α(t) +∫ t

aα(s)β(s)e

∫ t

sβ(τ) dτ ds.

(#2)(Existence and uniqueness theorem).Suppose I ⊆ IR1 is a (possibly unbounded) interval, O ⊆ IRn is open, andf : I × O → IRn is continuous.(a) (Lipschitz functions). (i) State what it means for f(t, ·) to be locallyLipschitz on O with Lipschitz constant independent of t ∈ [a, b] ⊆ I. (ii)What about globally Lipshitz? (iii) Show that if f(t, ·) is differentiable and(t, y) 7→ ∂

∂yf(t, y) is continuous [a, b] × O where [a, b] is a bounded interval,

then f(t, ·) is locally Lipshitz on O with constant independent of t ∈ [a, b].(iv) On what intervals [a, b] of I := (0,∞) is the function f(t, y) := t2y +√‖y‖ locally (resp. globally) Lipschitz independent of t ∈ [a, b]? (v) Answer

part (iv) for the function f(t, y) := ty2 with I = IR1.(b) (First-order initial value problems). Suppose y0 ∈ O and t0 ∈ I so that[t0, t0 + δ] ∈ I for some δ > 0. Consider the initial value problem

(IVP)

{y′(t) = f(t, y(t)) for all t ∈ [t0, t1)y(t0) = y0.

(i) State conditions on f for which there exists t1 > t0 with [t0, t1] ⊆ I sothat (IVP) has a unique solution. (ii) Show that solving (IVP) is equivalentto solving the integral equation

y(t) = y0 +∫ t

t0f(s, y(s)) ds.

(State precisely what is meant by equivalence.) (iii) Prove the result statedin part (i) two different ways, one by using a fixed point theorem and theother by using time discretizations. (iv) Suppose y(·) is a solution to (IVP)and its range is contained in some compact set K ⊂ O. Show that limt↑t1 y(t)exists. (v) Suppose O = IRn, I = [t0,∞), the assumptions of part (i) hold,and there exists constants c1 and c2 so that

‖f(t, y)‖ ≤ c1 + c2‖y‖ ∀y ∈ IRn, t ∈ [t0,∞). (1)

Show that for any y0 ∈ IRn, (IVP) has a solution y(·) defined on [t0,∞).Give an example for which all of the conditions hold except for the growthcondition (1), and for which the conclusion does not hold. (vi) Supposewe have the same assumptions as in part (v), K ⊆ IRn is compact, andt0 ≤ t1 < ∞. Show there exists a constant k (depending on K, t1 and theconstants in (1)) so that if y0 ∈ K and y(·) solves (IVP), then ‖y(t)‖ ≤ kfor all t ∈ [t0, t1]. (vii) Suppose we have the setup in part (v), and supposeK ⊆ IRn is compact, and t1 < ∞. Show that there exist a constant ω sothat if x0, y0 ∈ K and x(·), y(·) solve the ODE in (IVP) with x(t0) = x0 andy(t0) = y0, then

‖x(t)− y(t)‖ ≤ eω(t−t0)‖x0 − y0‖ ∀t ∈ [t0, t1].

(viii) Suppose we return to the setup in part (i). Let y(·) : [t0, t1) → Odenote a maximal solution to (IVP) in the sense of graph inclusion; that is,if y(·) : [t0, t2) → O is another solution, then t2 ≤ t1 and y(t) = y(t) for allt ∈ [t0, t2). Show that a maximal solution y(·) exists. (ix) Suppose y(·) is asin part (viii) and t1 < ∞. Show that for every compact subset K ⊂ O, thereexists t0 < t < t1 so that y(t) /∈ K for t ∈ [t, t1). (x) Suppose f(t, y) = f(y)(is autonomous) and is locally Lipschitz, f(0) = 0, and y0 6= 0. Show thatthe solution y(·) to (IVP) is never 0. (xi) Suppose again f is autonomousand satisfies the usual hypotheses asked for in part (i). Let φ(·, t0, y0) denotethe solution to (IVP). In terms of φ, what is the solution to the initial valueproblem of the form (IVP) but with f replaced by −f? (xii) Again assume f

is independent of t. Let ω+(y0) := {φ(t; t0, y0) : t ≥ t0} ⊆ O be the (positive)orbit, and ω(y0) := ω+(y0) ∪ {y1 : y0 ∈ ω+(y1)} be the (total) orbit of thesolution to (IVP). That is, ω+(y0) is the set of points that are in the rangeof a solution to (IVP) in positive time, and ω(y0) is the same but for positiveand negative time. Show that the positive and total orbits are independentof t0, and thus the notation ω+(y0), ω(y0) make sense. (xiii) Show that iftwo positive orbits intersect, then one is contained in the other. That is, ifω+(y0)∩ω+(y1) 6= ∅, then ω+(y0) ⊆ ω+(y1) or ω+(y1) ⊆ ω+(y0). (xiv) Showthat two total orbits intersect if and only if they are the same set. That is,ω(y0) ∩ ω(y1) 6= ∅ if and only if ω(y0) = ω(y1).

(c) (nth order scalar equations). Suppose g : I × O → IR1, and ~ξ =(ξ0, . . . , ξn−1)

Tr ∈ O. Consider the nth order scalar ODE

(ODEn)

{x(n)(t) = g(t, x(t), x′(t), . . . , x(n−1)) for all t ∈ [t0, t1)x(t0) = ξ0, x

′(t0) = ξ1, . . . , x(n−1 = ξ(n−1)

(i) Show that (ODEn) can be put in the form (IVP). (ii) State an existenceand uniqueness theorem for (ODEn).

Part II: Linear Theory.This part is devoted to linear problems, in which it is assumed that I = IR1,O = IRn, and the function f is given by f(t, y) = A(t)y, where A : I →Mn×n

(= the set of all n× n matrices) is continuous.(a) (Fundamental solutions) (i) Show that for every y0 ∈ IRn and −∞ <t0 < t1 < +∞, (IVP) has a unique solution. (ii) Suppose we are given a mapΦ : {(s, t) : s ≤ t} → Mn×n. Show that the following two sets of conditionsare equivalent:(I) There exists t0 ∈ IR so that Φ(·, ·) satisfies

(α) ddt

Φ(t0, t) = A(t)Φ(t0, t) ∀ t ≥ t0, and ddt

Φ(t0, t) = −A(t)Φ(t0, t) ∀ t ≤t0,

(β) Φ(t0, t0) = E (= the identity),

(γ) Φ(t0, t) = Φ(s, t)Φ(t0, s) ∀ s, t ∈ IR.

(II) Φ(·, ·) satisfies

(α) ddt

Φ(s, t) = A(t)Φ(s, t) ∀ s ∈ IR and t ≥ s,

(β) Φ(s, s) = E ∀ s ∈ IR,

(γ) Φ(s, t) = Φ(s, t)−1 ∀ s, t ∈ IR.

A map Φ satisfying either (and therefore both) (I) or (II) is called afundamental solution. (iii) Show that fundamental solutions always exist,and can be chosen with Φ(t0, t0) = E, the identity matrix. (iv) Show thatif Ψ(·, ·) is any other fundamental solution with Ψ(t1, t1) = E, then thereexists a nonsingular matrix M so that Φ(t0, t) = Ψ(t1, t)M for all t ∈ IR1.(v) Let Φ be given as in part (iii), and show that the solution to (IVP)is given by y(t) = Φ(t0, t)y0 (which is a special case of the variations ofconstants formula. (vi) What is the variations of constants formula? Givenany fundamental solution Ψ, the variation of constants formula represents thesolution to (IVP); what is this formula? (This reduces to the representationformula in part (v) if Ψ(t0, t0) = E). (vii) Show that if each column of a mapΦ : IR1 × IR1 →Mn×n is a solution to the ODE in (IVP), and det Φ(t0, t) 6=0 for one choice of (t0, t), then det Φ(s, t) 6= 0 for all (s, t) ∈ IR1 × IR1.(viii) Define the Wronskian of a linear system, and characterize fundamentalsolutions in terms of it. (ix) Show that if Φ(s, t) is fundamental for the systemdefined by the matrices A(t), then Ψ(s, t) := Φ(t, s) is fundamental for thesystem defined by −A(t), and that Ψ(s, t) = Φ(s, t)−1 for all s ≤ t.(b) (Some linear algebra used in ODE theory) (i) Suppose A is an n di-mensional matrix. Define the exponential exp (A), and illustrate why itis well defined (that is, why the series converges). (ii) For given n × nmatrices A and B, when does exp(A + B) = exp (A)exp(B)? (iii) Findexp (Ai) for the matrices Ai, i = 1, . . . , 4 given below. (iv) Show thatexp (M−1AM) = M−1exp (A)M for any nonsingular matrix M . (v) Showthat exp (A) is always nonsingular, and its inverse is given by exp (−A).(c) (Autonomous linear systems) Now suppose f(t, y) = A(t)y = Ay for allt, where A is a given matrix. (i) Show that Φ(t0, t) := exp (A(t − t0)) is afundamental solution associated to (IVP). (ii) Find the general solution of(IVP) for each system with A = Ai, i = 4, . . . , 7, where the matrices aregiven below. (iii) Sketch the phase portrait of each system from part (ii),(iv) Let ω ∈ IR1 be larger than the real part of any eigenvalue of A. Showthat for any r > 0, there exists a constant k > 0 so that any solution y(·) of(IVP) with ‖y0‖ ≤ r satisfies ‖y(t)‖ ≤ keωt.

A1 =

2 1 0 00 2 1 00 0 2 00 0 0 1

A2 =

3 0 0 00 −2 0 00 0 2 10 0 0 2

A3 =

2t t 0 00 2t t 00 0 2t t0 0 0 2t

A4 =

(0 11 0

)A5 =

(3 113 1

)A6 =

(2 10 2

)A7 =

(1 1−4 1

)

Part III: Stability Theory.Suppose f : IRn → IRn is globally Lipschitz with f(0) = 0.

(a) (Definitions) State precisely the definitions for the equilibrium point 0to be (i) stable, and (ii) asymptotically stable.

(b) (Linear systems) (i) Suppose f(t, y) = Ay, where A is an n× n matrix.Give sufficient conditions for stability and asymptotic stability in terms ofthe eigenvalues of A, and provide proofs. (ii) Suppose f(t, y) = A(t)y whereA : [0,∞) → Mn×n is continuous and bounded. Show that the system isstable if and only if every solution to (IVP) is bounded (and only dependson ‖y0‖).(c) For nonlinear, autonomous f that is locally Lipschitz, give a sufficientcondition for each of (i) and (ii) in terms of a Lyapunov function. (Of courseyou need to define a Lyapunov function.) Prove these results.