Solve the following equations using the Laplace transform. (a) y" - 2y'+2y=et; y(0) = 0, y'(0) = 1 File Preview s²Y – sy(0) — y' (0) — 2sY+2y(0) + 2Y = 1 S+1 1 s2Y 1-2sY+2Y = s+1 1 (s2 2s+2)Y= +1 s+1 s+2 A Bs + C Y(s): = + (s+1) (s2 2s+2) S+1 s²-2s+2 To determine the coefficients, the RHS is As² 2As+2A + Bs² + Bs + Cs + C (s+1) (s² - 2s+2) (A+B)s²+(-2A + B + C)s + 2A+ C (s+1) (s2 2s+2) By comparing the numerator with s +2 we have A + B = 0, -2A + B + C = 1 and 2A+ C = 2. Therefore B = -A and -2A- A+ C = −3A+ C = 1. Combining this with 2AC 2 we have A = 1/5, B = -1/5 and C = 8/5. 1 1 1 1 Y(s) = + = - 5s +1 (s - 1)²+1 5s +1 1 S-1 5 (s - 1)²+1 By checking the table of Laplace transform 1 7 et y(t) = e² - 1½ e' cost + e' sir e² sint 7 1 + 5 (s-1)²+1
Solve the following equations using the Laplace transform. (a) y" - 2y'+2y=et; y(0) = 0, y'(0) = 1 File Preview s²Y – sy(0) — y' (0) — 2sY+2y(0) + 2Y = 1 S+1 1 s2Y 1-2sY+2Y = s+1 1 (s2 2s+2)Y= +1 s+1 s+2 A Bs + C Y(s): = + (s+1) (s2 2s+2) S+1 s²-2s+2 To determine the coefficients, the RHS is As² 2As+2A + Bs² + Bs + Cs + C (s+1) (s² - 2s+2) (A+B)s²+(-2A + B + C)s + 2A+ C (s+1) (s2 2s+2) By comparing the numerator with s +2 we have A + B = 0, -2A + B + C = 1 and 2A+ C = 2. Therefore B = -A and -2A- A+ C = −3A+ C = 1. Combining this with 2AC 2 we have A = 1/5, B = -1/5 and C = 8/5. 1 1 1 1 Y(s) = + = - 5s +1 (s - 1)²+1 5s +1 1 S-1 5 (s - 1)²+1 By checking the table of Laplace transform 1 7 et y(t) = e² - 1½ e' cost + e' sir e² sint 7 1 + 5 (s-1)²+1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Can you explain the algebra to get A B C
Transcribed Image Text:Solve the following equations using the Laplace transform.
(a) y" - 2y'+2y=et; y(0) = 0, y'(0) = 1
File Preview
s²Y – sy(0) — y' (0) — 2sY+2y(0) + 2Y =
1
S+1
1
s2Y 1-2sY+2Y =
s+1
1
(s2 2s+2)Y=
+1
s+1
s+2
A
Bs + C
Y(s):
=
+
(s+1) (s2 2s+2)
S+1 s²-2s+2
To determine the coefficients, the RHS is
As² 2As+2A + Bs² + Bs + Cs + C
(s+1) (s² - 2s+2)
(A+B)s²+(-2A + B + C)s + 2A+ C
(s+1) (s2 2s+2)
By comparing the numerator with s +2 we have A + B = 0, -2A + B + C = 1 and
2A+ C = 2. Therefore B = -A and -2A- A+ C = −3A+ C = 1. Combining this with
2AC 2 we have A = 1/5, B = -1/5 and C = 8/5.
1 1
1 1
Y(s)
=
+
=
-
5s +1 (s - 1)²+1
5s +1
1
S-1
5 (s - 1)²+1
By checking the table of Laplace transform
1
7
et
y(t) = e² - 1½ e' cost + e' sir
e² sint
7
1
+
5 (s-1)²+1
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