# Homework Assignment 16 in Differential Equations, MATH308-Fall 2015

```Homework Assignment 16 in Differential Equations, MATH308-Fall 2015
This homework is not for the submission
but it is highly recommended to solve it because the material of it is in the final test
Topics covered : step function,Laplace transform of discontinuous functions and differential equations
with discontinuous forcing function(corresponds to sections 6.3, 6.4), convolution interal (section 6.6)
1. Find the Laplace transform of the function


t &lt; 2,
5t
2
f (t) = t + t − 2 2 ≤ t &lt; 6,


1 + 4t
6 ≤ t.
π
2. Find the inverse Laplace transform of the function
e− 2 s (3s2 − 2s + 1)
.
(s + 1)2 (2s2 + 12s + 26)
3. Use the convolution theorem to find the inverse Laplace transform of the given function:
s2
(s2 + 9)(s2 + 25)
4. Find the solution of the initial value problem y 00 + 8y 0 + 25y = g(t); y(0) = 1, y 0 (0) = −2, where
g(t) =


2 cos 2t,
3π
2
3π
t≥
.
2
0≤t&lt;

1 − 3 sin 2t,
5. (a) Express the solution of the given initial value problem in terms of a convolution integral:
y 00 − 10y 0 + 29y = g(t),
y(0) = 0, y 0 (0) = −2.
(1)
(b) Find the solution of the same initial value problem (1) using the method of variation of