18.034, Honors Differential Equations
Prof. Jason Starr
Rec. Suggestions.
4/7/04
1. Do 1 or 2 more IVP’s by Laplace transform including computation of partial fractions, e.g
y”+y = cos(2t)
Y (s) =
y(0) = 0
y’(0) = 1
1
1 s
1 s
+
−
2
s +1 3 s +1 3 s2 + 4
2
y(t ) = sin(t ) +
1
1
cos(2t ) − cos(2t ) .
3
3
2. Use periodic function rule to deduce,
2π
∫e
− st
0
(
2π
)
s
cos(t )dt = 2
1 − e − 2πs ,
s +1
∫e
− st
sin(t )dt =
0
(
1
1 − e − 2πs
s +1
2
)
You might mention that taking the power series in s of each side gives closed formulas,
2π
e.g.
∫
t k + 1 sin(t )dt = (k + 1)!
0
⎡z⎤
⎢ ⎥
⎣υ ⎦
(−1) l + 1 (2π ) k + 1 − 2l
(k + 1 − 2l )!
l =0
∑
This can be used to compute Fourier coefficients of tk+1.
3. Use L to prove S(t)ta ∗ S(t)tb =
a! b!
S(t) ta+b +1
(a + b + 1)!
S(t) ∗ cos(t) = S(t)sin(t)
4. Consider p(s) = s 2 + As - B . In the 3 cases that
(i)
p(s) = (s - r1 )(s - r2 ) , (ii)
p(s) = (s - r)2 ,
(iii)
p(s) = (s - α )2 + ß 2 ,
⎡ 1 ⎤
compute L− 1 ⎢
⎥ . (i.e. first row of Table 5.4.1).
⎣ p(s) ⎦
18.034, Honors Differential Equations
Prof. Jason Starr
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