Math 325: Differential Equations	Assignment 2		Fall 1997

Names (up to 3 students):

Section:

Initialization

<<Calculus`LaplaceTransform`

Problem 1

a) Find the Laplace transforms of the functions t*E^(3t) and t^2 Sin[5t] by using Integrate[]. Compare the result with 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 partial fraction decompositions (use Apart[]) and the tables on p.300 of the textbook. Compare the result with InverseLaplaceTransform[].

Solution

Problem 2

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

Solution

Problem 3

Solve the initial value problems using the Laplace transform and plot the solutions:
a) y''''[t] + 2y'''[t] + 2y''[t] + 2y'[t] + y[t] == 10 UnitStep[t-5],
	y[0] == 0, y'[0] == 0, y''[0]==0, y'''[0]==0
b) y''''[t] + 8y''[t] + 16y[t] == DiracDelta[t-Pi] - DiracDelta[t-4Pi],
	y[0] == 0, y'[0] == 0, y''[0]==0, y'''[0]==0

Solution

Problem 4

Consider the initial value problem
	y''[t] + 1/3 y'[t] + 4y[t] == f[k,t], y[0] == 0, y'[0] == 0
where f[k_,t_] = 1/(2k) if 4-k <= t < 4+k and is 0 otherwise.
Solve the equation using Laplace transforms by writing f[k,t] in terms of the unit step function.
Superimpose plots of f[k,t] for k == 3/2, 1, and 1/2 and then superimpose plots of the solutions to the initial value problem for the same values of k. Try to determine the limiting the solution as k approaches 0.

Solution

Problem 5

Consider the initial value problem
	y''[t] + h y'[t] + y[t] == k*DiracDelta[t-1],
	y[0] == 0, y'[0] == 0
where k is the magnitude of an impulse at t==1 and h is the damping coefficient (or resistance).
a) Let h==0.5. Find the value of k for which the response has a peak value of 2. Call this value k0.
b) Repeat part a) for h==0.25.
c) Determine how k0 varies as h decreases. What is the value of k0 when h==0?

Solution

Problem 6

a) Use the Convolution Theorem 6.6.1 (p.326) to express the inverse Laplace transform of the function
H[s_] = s/((s+1)^2(s^2+9)) in terms of an integral. Evaluate this integral with Integrate[] and compare the final result with InverseLaplaceTransform[H[s],s,t].
b) Express the solution of the initial value problem
	y''''[t] + 10y''[t] + 9y[t] == g[t],
	y[0]==2, y'[0]==1, y''[0]==0, y'''[0]==0
in terms of a convolution integral. (Use LaplaceTransform[] and InverseLaplaceTransform[]).
c) Use the result of b) to generate solutions to the initial value problem when g[t_] is Cos[t],  E^t, and t^2. (You should only need to make a substitution and carry out the integration).

Solution

Problem 7

a)  Find the eigenvalues of the matrix
	A = {{ 7,-3, 3},
	     {-1, 5, 1},
	     { 2,-2, 8}}
by using Det[]. Then find the eigenvectors corresponding to each eigenvalue r by solving the appropriate linear system.
Compare your results to the output of Eigensystem[].
b) Find a matrix T such that Inverse[T].A.T == d where d is a diagonal matrix (see p. 362). Verify your choice of T by calculating Inverse[T].A.T.

Solution

Problem 8

Consider the vector functions
	X1[t_] = { t , t^2, t^3};
	X2[t_] = { t , t^2, 1  };
	X3[t_] = {t^2,  1 , t  };
a) For what values of t are the vectors X1[t], X2[t], X3[t] linearly independent?
b) On what intervals are the vector functions X1[t], X2[t], X3[t] linearly independent?
c) What conclusion can be draw about the coefficients in a system of homogeneous differential equations satisfied by
    X1, X2, X3?
d) Find such a system and verify the conclusions of part c).

Solution