Page 138 - Advanced engineering mathematics

118 CHAPTER 3 The Laplace Transform

SECTION 3.7 PROBLEMS

Solve each of the following problems using the Laplace 7. y + 8ty = 0; y(0) = 4, y (0) = 0
transform.

8. y − 4ty + 4y = 0; y(0) = 0, y (0) = 10
2
1. t y − 2y = 2 Hint: First set u = 1/t. 9. y − 8ty + 16y = 3; y(0) = y (0) = 0

2. y + 4ty − 4y = 0; y(0) = 0, y (0) =−7 10. (1 − t)y + ty − y = 0; y(0) = 3, y (0) =−1

3. y − 16ty + 32y = 0; y(0) = y (0) = 0 11. Review the derivation of the solution of Bessel’s equa-

4. y + 8ty − 8y = 0; y(0) = 0, y (0) =−4 tion of order n for n a positive integer. Are any steps

taken that would prevent n being an arbitrary posi-
5. ty + (t − 1)y + y = 0; y(0) = 0 tive number, not necessarily an integer? Could n be

6. y + 2ty − 4y = 6; y(0) = y (0) = 0 negative?

Appendix on Partial Fractions Decompositions
Partial fractions decomposition is an algebraic manipulation designed to write a quotient
P(x)/Q(x) of polynomials as a sum of simpler quotients, where simpler will be defined by
the process.
Let P have degree m and let Q have degree n and assume that n > m. If this is not the case,
divide Q into P. Assume that P and Q have no common roots, and that Q has been completely
factored into linear and/or irreducible quadratic factors. A factor is irreducible quadratic if it
is second degree with complex roots, hence it cannot be factored into linear factors with real
2
coefficients. An example of an irreducible quadratic factor is x + 4.
The partial fractions decomposition consisting of writing P(x)/Q(x) as a sum S(x) of
simpler quotients is given in the following rules.
2
1. If x − a is a factor of Q(x) but (x − a) is not, then include in S(x) a term of the form
A
.
x − a
k
2. If (x − a) is a factor of Q(x) with k > 1but (x − a) k+1 is not a factor, then include in
S(x) a sum of terms of the form
B 1 B 2 B k
+ + ··· + .
x − a (x − a) 2 (x − a) k
2
3. If ax + bx + c is an irreducible quadratic factor of Q(x) but no higher power is a factor
of Q(x), then include in S(x) a term of the form
Cx + D
.
ax + bx + c
2
2
2
k
4. If (ax + bx + c) is a product of irreducible factors of Q(x) but (ax + bx + c) k+1 is not
a factor of Q(x), then include in S(x) a sum of terms of the form
C 1 x + D 1 C 2 x + D 2 C k x + D k
+ + ··· + .
2
2
ax + bx + c (ax + bx + c) 2 (ax + bx + c) k
2
When each factor of Q(x) has contributed one or more terms to S(x) according to these
rules, we have an expression of the form
P(x)
= S(x),
Q(x)
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