Math 325: Differential Equations	Assignment 	Fall 1998

Name:

Section:

Professor:

This assignment is due Monday, Nov 23, 1998.   Print out your answer to the
problems.  This assignment will be counted as one homework.

Initialization

<<Calculus`LaplaceTransform`

Problem 1

Find the indefinite integral of the following functions
	E^t Cos[t]^3 Sin[t]^5;      (t-1)^10 Log[t+1]

Solution

Problem 2

Verify that the functions
	y1[t_] = 1;      y2[t_] = t;
	y3[t_] = E^(-t); y4[t_] = t*E^(-t);
are solutions of the differential equation
	y''''[t] + 2y'''[t] + y''[t] == 0
and calculate their Wronskian.

Solution

Problem 3

Show that the Wronskian of Cos[t], Cos[t]^3, Cos[t]^5, and Cos[5t] is 0.
Establish this result without direct evaluation of the Wronskian.

Solution

Problem 4

Find all the 10th roots of 1024, including complex numbers. Convert these to points in the plane and plot them.

Solution

Problem 5

Solve the Differential Equation 
 y'''[t] + 17y''[t] + 22y'[t] -40y[t] == 0
with the initial conditions:
	y[0] == 1, y'[0] == 99, y''[0]==999, 
using DSolve[].

Solution

Problem 6

Find the solutions to the following differential equation:
	y''''[t] + 2y'''[t] + 5y''[t] + 8y'[t] + 4y[t]==Sin[2t] + t*E^(-t)
with the initial conditions
	y[0] == y0, y'[0] == 0, y''[0] == 0, y'''[0] == 0
Superimpose plots of the particular solutions for several values of y0. Explain why all these solutions seem to pass through the t-axis at the same points and at regular intervals for large values of t. Try to determine the length of this common interval.

Solution

Problem 7

a) Find the Laplace transforms of the functions t*E^(3t) and t^2 Sin[5t] by using  LaplaceTransform[].
b) Find the inverse Laplace transform of the functions (s-1)/(s^2-4) and
(s^2 + s + 5)/(s^3 - 4s^2 + 5s) by using InverseLaplaceTransform[].

Solution

Problem 8

Solve the initial value problems using the Laplace transform. Compare the answers with the results given by DSolve[] and plot the solutions:
 y''[t] + 9y[t] == Cos[2t], y[0] == 1, y'[0] == 0

Solution

Problem 9

For each of the matrices A below, find the fundamental matrix Phi[t] such that Phi[0] is the identity matrix for the system X'[t] == A.X[t].  Then use Phi[t] to find a solution satisfying the given initial condition.

a)  A = {{-7,-8, 2},
        { 9,11,-3},
        { 9, 8, 0}};
  X[0]=={3,-1,1};

b)  A = {{-4,-2,-1},
        { 2, 6,-3},
        { 2, 4,-1}};
  X[0]=={1,1,-2};
 
c)  A = {{  5, 3,-1},
        {-16,-6,-1},
        { -4,-3, 2}};
  X[0]=={1,1,1};

Solution

Problem 10

Use the method of variation of parameters to find the general solution of the system X'[t] == A.X[t] + G[t] for each of the matrices A and vector functions G[t] below. . Then find the solution that satisfies the given initial condition.

(see Problem 9a)
   A = {{-7,-8, 2},
        { 9,11,-3},
        { 9, 8, 0}};
  G[t_] = {t^2,t,1};
  X[0]=={3,-1,1};

Solution