Summary Differential Equations with Boundary-Value Problems, Exercise 7.1, 7 to 10

Dennis G. Zill - Differential Equations with
Boundary-Value Problems, Exercise 7.1. From 11 to 18
Leonardo J. Perez Gomez
14 de agosto de 2022


1. Let the figure 1


2.00



1.75



1.50



1.25
f(t)




1.00



0.75



0.50



0.25



0.00


0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
t




Figura 1: Problem 7



0 0≤t<1
∴f (t) =
f1 (t) t≥1
If f1 (t) = t

0 0≤t<1
∴f (t) =
t t≥1
Z ∞ Z 1 Z ∞
∴ L {f (t)} = −st
e f (t)dt = −st
e (0)dt + e−st (t)dt
Z0 ∞ 0 1

= e−st (t)dt
1




1

, e−st (t)dt
R
1.

dv = e−st dt e−st t e−st
Z Z
u=t −st
−st ∴ e dt = − − − dt
du = dt v = − e s s s
e−st t 1 e−st t 1
Z  −st 
−st e
=− + e dt = − + −
s s s s s
−st −st
 
e t e t 1
=− − 2 = e−st − − 2
s s s s

  b
t 1
∴ L {f (t)} = lı́m e − − 2 −st
b→∞ s s 1
 
−s 1 1
=e − − 2
s s

2. Let the figure 2


4.0



3.5



3.0



2.5
f(t)




2.0



1.5



1.0



0.5



0.0


0.0 0.5 1.0 1.5 2.0 2.5 3.0
t




Figura 2: Problem 8



0 0≤t<1
∴ f (t) =
f1 (t) t≥1
If f1 (t) = 2t − 2

0 0≤t<1
∴ f (t) =
2t − 2 t≥1
Z 1 Z ∞ Z ∞
∴ L {f (t)} = e −st
(0)dt + e−st
(2(t − 1))dt = e−st (2(t − 1))dt
0 1 1




2