Sample Problems for the Midterm ExamUCLA Math 135, Winter 2015, Professor David Levermore

(1) Give the interval of definition for the solution of the initial-value problem

d3x

dt3+

cos(3t)

4− tdx

dt=

e−2t

1 + t, x(2) = x′(2) = x′′(2) = 0 .

(2) Suppose that Y1(t) and Y2(t) are solutions of the differential equation

y′′ + 2y′ + (1 + t2)y = 0 .

Suppose you know that W [Y1, Y2](0) = 5. What is W [Y1, Y2](t)?

(3) Show that the functions Y1(t) = cos(t), Y1(t) = sin(t), and Y3(t) = 1 are linearlyindependent.

(4) Let L be a linear ordinary differential operator with constant coefficients. Supposethat all the roots of its characteristic polynomial (listed with their multiplicities) are−2 + i3, −2− i3, i7, i7, −i7, −i7, 5, 5, 5, −3, 0, 0.(a) Give the order of L.(b) Give a general real solution of the homogeneous equation Ly = 0.

(5) Let D =d

dt. Solve each of the following initial-value problems.

(a) D2y + 4Dy + 4y = 0 , y(0) = 1 , y′(0) = 0 .

(b) D2y + 9y = 20et , y(0) = 0 , y′(0) = 0 .

(6) The functions cos(2t) and sin(2t) are a fundamental set of solutions to y′′ + 4y = 0.Find the solution Y (t) to the general initial-value problem

y′′ + 4y = 0 , y(0) = y0 , y′(0) = y1 .

(7) Let D =d

dt. Give a general real solution for each of the following equations.

(a) D2y + 4Dy + 5y = 3 cos(2t) .

(b) D2y − y = t et .

(c) D2y − y =1

1 + et.

(8) Let D =d

dt. Consider the equation

Ly = D2y − 6Dy + 25y = et2

.

(a) Compute the Green function g(t) associated with L.(b) Use the Green function to express a particular solution YP (t) in terms of definite

integrals.

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(9) Compute the Green function g(t) associated with the differential operator

L = D2 − 4D + 4 , where D =d

dt.

(10) Solve the initial-value problem

w′′ − 4w′ + 4w =2e2t

1 + t2, w(0) = w′(0) = 0 .

(11) Compute the Laplace transform of f(t) = t e3t from its definition.

(12) Compute the Laplace transform of f(t) = u(t− 5) e2t from its definition.(Here u is the unit step function.)

(13) Compute the Laplace transform of cos(t)δ(t− π) from its definition.(Here δ is the unit impulse.)

(14) Find the Laplace transform Y (s) of the solution y(t) of the initial-value problem

y′′ + 4y′ + 13y = f(t) , y(0) = 4 , y′(0) = 1 ,

where

f(t) =

{cos(t) for 0 ≤ t < 2π ,

t− 2π for t ≥ 2π .

You may refer to the table on the last page. DO NOT take the inverse Laplacetransform to find y(t), just solve for Y (s)!

(15) Find the inverse Laplace transforms of the following functions.You may refer to the table on the last page.

(a) F (s) =2

(s+ 5)2,

(b) F (s) =3s

s2 − s− 6,

(c) F (s) =(s− 2)e−3s

s2 − 4s+ 5.

(16) Find the inverse Laplace transform of Y (s) =3

(s2 − 4s+ 5)(s2 − 4s+ 8).

You may refer to the table on the last page.

(17) Let f(t) be continuous over [0,∞) and be of exponential order α as t→∞ for someα ≥ 0. Define h(t) by

h(t) =

∫ t

0

f(t′) dt′ for every t ∈ [0,∞) .

(a) Show that h(t) is of exponential order α as t→∞.(b) Express H(s) = L[h](s) in terms of F (s) = L[f ](s).

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(18) Solve the initial-value problem

y′′ + y′ − 6y = 7δ(t− 4) , y(0) = 0 , y′(0) = 5

Here δ is the unit impulse. You may refer to the table on the last page.

(19) Let f(t) = cos(2t) and g(t) = e3t. Compute f ∗ g(t).You may refer to the table on the last page.

(20) Solve the integral equation e−t = y(t) + 2 cos ∗y(t).You may refer to the table on the last page.

(21) Consider the initial-value problem

y′ = 3y23 , y(0) = −8 .

(a) What is the largest time interval over which this problem has a unique solution?(b) Find the unique solution over this time interval.(c) Find two solutions that extend beyond this time interval.

(22) Let f(t, y) = cos(t)y + y3. For each r > 0 consider the set

Sr ={

(t, y) : t ∈ R , y ∈ [−r, r]}.

(a) Find Mr, the smallest upper bound on |f(t, y)| over Sr.(b) Find Lr, the smallest Lipschitz constant for the y variable of f(t, y) over Sr.

(23) Consider the initial-value problem

y′ = cos(t)y + y3 , y(0) = 1 .

(a) If the initial Picard iterate is y0(t) = 1, compute the first Picard iterate y1(t).(b) Use the Picard Theorem to show that this problem has a unique solution over a

time interval [0, h] for some h > 0.(c) Show that the solution of this problem blows up at a finite positive time.

(24) Transform the equation v′′′+t2v′−3v3 = sinh(2t) into a first-order system of ordinarydifferential equations.

A Short Table of Laplace Transforms

L[tneat](s) =n!

(s− a)n+1for s > a .

L[eat cos(bt)](s) =s− a

(s− a)2 + b2for s > a .

L[eat sin(bt)](s) =b

(s− a)2 + b2for s > a .

L[tnj(t)](s) = (−1)nJ (n)(s) where J(s) = L[j(t)](s) .

L[eatj(t)](s) = J(s− a) where J(s) = L[j(t)](s) .

L[u(t− c)j(t− c)](s) = e−csJ(s) where J(s) = L[j(t)](s)

and u is the unit step function .