hw 3 In the following exercises, you need to use Laplace transform

1. hw 3

In the following exercises, you need to use Laplace transform. Let f(t) be a smooth ¹ function on [0,∞) whose Laplace transformL(f)(s) exists on a domainDfors.Fors∈D, denote F(s) =L(f)(s),i.e.,

F(s) = Z ∞

0

e^−stf(t)dt.

Let us denotef^(k)(t) thek-th derivative off and setf⁽⁰⁾(t) =f(t).Suppose that

Nlim→∞f^(k)(N)e^−sN = 0, k≥0.

(1) For alln≥1,show that

L(f⁽ⁿ⁾)(s) =sⁿF(s)−X

k=0

s^n−k−1f^(k)(0) by induction.

(2) Solve fory=f(t) in the following initial value problem using Laplace transform and the table of Laplace transform.

(a) y⁰+ 3y= 0 withy(0) = 1.

(b) y⁰⁰−3y⁰+ 2y= 0 withy(0) = 1 andy⁰(0) =−1.

(c) y⁰⁰−2y⁰+y= 0 withy(0) = 3 andy⁰(0) = 2.

(d) y⁰⁰+ 4y= 0 withy(0) = 2 andy⁰(0) =−1.

(3) LetF(s) = 3s+ 1 s³−s²+s−1.

(a) Find the partial fraction expansion ofF(s) : 3s+ 1

s³−s²+s−1 =A 1

s−1 +B s

s²+ 1+C 1 s²+ 1. (b) Solve forf(t) using the table of Laplace transform.

In the following exercises, you need to use Gamma function and Beta function.

(1) Using Beta function to compute Z ∞

0

e^−x2dx.

(2) Find the following improper integrals.

(a) Z ∞

0

e^−10tt²dt

(b) Z ∞

0

e^−t2t¹⁰dt.

(3) Letn≥1.Supposex >0.Show that 1

Γ(x) Z ∞

0

t^x−1 e^−t−e^−(n+1)t 1−e^−t

! dt=

n

X

k=1

1 k^x. (4) Suppose thatp, q >0.Show that

Z 1

−1

(1 +x)^p−1(1−x)^q−1dx= 2^p+q−1B(p, q).

(5) Compute the following integrals.

(a) Z ∞

0

x³

(1 +x²)¹⁰dx.

1In other words,f⁽ⁿ⁾(t) exists and continuous on [0,∞) for alln≥1.

1

2

(b) Z 1

0

(1−x²)¹⁰x⁵dx.

(c) Z ∞

0

t¹⁰ (1 +t)¹⁰⁰dt.

(d) Z 3

2

(t−2)⁵(t−3)¹⁰dt.

(e) Z π/2

0

sin⁸xdx.

(f) Z π/2

0

cos¹¹xsin¹³xdx.