Ordinary Differential Equations
MATH 308 - 523
A. Bonito
March 31
Spring 2015
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Quiz 4
• 5 minute individual quiz;
• Answer the questions in the space provided. If you run out of space, continue
onto the back of the page. Additional space is provided at the end;
• Show and explain all work;
• Underline the answer of each steps;
• The use of books, personal notes, calculator, cellphone, laptop, and communication with others is forbidden;
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Exercise 1
100%
Find the solution of
y 00 + y = g(t),
where
g(t) :=
y(0) = y 0 (0) = 0,
0
2
t < 3π,
t > 3π.
Ordinary Differential Equations
MATH 308 - 523
A. Bonito
March 31
Spring 2015
Quiz 4: solutions
Exercise 1
100%
We write the right and side using a step function
g(t) = 2u3π (t),
so that taking the Laplace tranform yields
(s2 + 1)Y =
2 −3πs
e
,
s
where Y = L(y). As a consequence, we obtain
Y =
2
e−3πs .
+ 1)
s(s2
We first find the Laplace transform inverse of F (s) :=
2
s(s2 +1)
using simple fractions
2
A Bs + C
2
2s
= + 2
= − 2
.
s(s2 + 1)
s
s +1
s s +1
Hence,
L−1 (F )(t) = 2 − 2 cos(t).
Then, using the formula
L−1 (F (s)e−cs ) = L−1 (F )(t − c)uc (t)
we arrive at
y(t) = (2 − 2 cos(t − 3π))u3π (t) = (2 + 2 cos(t))u3π (t).