Lecture 1 Sujin Khomrutai – 1 / 23
Method of Applied Math
Lecture 5: Laplace Transform
Sujin Khomrutai, Ph.D.
Introduction
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 2 / 23
• The Laplace Transform is a very efficient technique for solving Initial Value Problems (IVPs).
• This is a transformation technique:
IVP
⇓
Algebraic equations
⇓
Solving algebraic equations
⇓
Solutions to the IVPs
Initial Value Problems
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 3 / 23
Definition An Initial Value Problem (IVP) is a differential equation that is given with some conditions involving the function and some of its derivatives at a certain initial time.
• The prescribed values are called initial conditions.
• In practices, these values can be observed or measured by ex- periment.
Example 1
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 4 / 23
EX. The switch is on at time t = 2 (from B to A), where the
capacitor contained no charges before that time. After 1 min, the switch is turning back. Find the voltage drop Eout at any time t.
Example 2
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 5 / 23
EX. The spring-mass system is start from equilibrium at t = 0.
Then the mass start to move at the initial velocity of 2 (m/s) to the right and is exerted by an external force f(t) = 2 sin 5t. Find the movement y1 of the mass where we are given the data that k1 = k2 = 2, M = 5, m = 1, and c1 = 0.
Initial Value Problems
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 6 / 23
ODE of order n
any(n) + an−1y(n−1) + · · · + a1y′ + a0y = f(t) standard initial conditions are
y(t0) = A0, y′(t0) = A1, . . . , y(n−1)(t0) = An−1. Usually t0 = 0.
Initial Value Problems
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 7 / 23
EX. The ODE
y′′ − 3y′ + 2y = et with the initial conditions
y(0) = −2, y′(0) = 1 is an initial value problem.
Initial Value Problems
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 8 / 23
EX. The ODE
y′′′ − 3y′′ + y′ + 5y = tsin t with the initial conditions
y(1) = 0, y′(1) = 10, y′′(1) = 3 is an initial value problem.
Laplace Transform
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 9 / 23
Definition. Given a function f(t) whose domain contains [0, ∞), we define a funciton F(s) by
F(s) =
Z ∞
0
e−stf(t) dt
The domain of F is any s such that the above integral converges.
F(s) is called the Lapalce transform of f(t) and is denoted by F(s) = L[f(t)]
Example 3
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 10 / 23
EX. Find the Laplace transform of
f(t) = eat, g(t) = sin(at)
where a is a real number. Also, if a > −1 find the Laplace transform of
h(t) = ta.
Inverse Laplace Transform
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 11 / 23
Definition. Given a function F(s), a function f(t) such that L[f(t)] = F(s) is called Inverse Laplace transform of F(s) and is denoted by
f(t) = L−1[F(s)].
The inverse Laplace transform function f(t) is unique provided it is continuous. It can be easily found from the formulas of Laplace transform.
Example 4
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 12 / 23
EX. Find the inverse Laplace transform of F(t) = 1
s − a, G(t) = a
s2 + a2.
Table
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 13 / 23
f(t) F(s)
1 1
s (s > 0) tn (n = 1,2, . . .) n!
sn+1 (s > 0) ta (a > −1) Γ(a + 1)
sa+1 (s > 0)
eat 1
s − a (s > a)
sin(at) a
s2 + a2 (s > 0)
cos(at) s
s2 + a2 (s > 0)
Linearity Property
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 14 / 23
Theorem. Let L[f(t)] = F(s), L[g(t)] = G(s) and a, b are constants. Then
L[af(t) + bg(t)] = aF(s) + bG(s) L−1[aF(s) + bG(s)] = af(t) + bg(t) that is
L[af(t) + bg(t)] = aL[f(t)] + bL[g(t)]
L−1[aF(s) + bG(s)] = aL−1[F(s)] + bL−1[G(s)]
Linearity Property
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 15 / 23
In general,
L[af(t) + bg(t) + · · · ] = aL[f(t)] + bL[g(t)] + · · ·
L−1[aF(s) + bG(s) + · · · ] = aL−1[F(s)] + bL−1[G(s)] + · · · Proof. By the linearity property for integral, we have
L[af(t) + bg(t)] =
Z ∞
0
e−st (af(t) + bg(t)) dt
= a
Z ∞
0
e−stf(t)dt + b
Z ∞
0
e−stg(t) dt
= aL[f(t)] + bL[g(t)]
Example 5
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 16 / 23
EX. Find the Laplace transform of f(t) = cos 2t + 4t + 7e−2t.
Example 6
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 17 / 23
EX. Find the inverse Laplace transform of F(s) = 1
(s + 1)(s − 2).
Derivation Property
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 18 / 23
Theorem. Let L[f(t)] = F(s). Then L[f′(t)] = sF(s) − f(0)
and
L[f′′(t)] = s2F(s) − sf(0) − f′(0).
Generally,
L[f(n)(t)] = snF(s) − sn−1f(0) − · · · − sf(n−2)(0)
− f(n−1)(0)
Derivation Property
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 19 / 23
Proof. By the definition and the integration by parts, we have L[f′(t)] =
Z ∞
0
e−stf′(t) dt
= e−stf(t)
∞
0 −
Z ∞
0
(−se−st)f(t) dt
= −f(0) + sF(s)
The remaining identity can be proved in the same way.
Example 7
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 20 / 23
EX. Let y(0) = −2, y′(0) = 1. Find the Laplace transform of y′(t), y′′(t),
in terms of Y (s) = L[y(t)].
Example 8
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 21 / 23
EX. Assume that y(t) is the solution of the IVP
y′′ + 3y′ − 2y = sin(3t), y(0) = −1, y′(0) = 0. Calculate the Laplace transform Y (s) = L[y(t)].
Example 9
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 22 / 23
EX. Solve the IVP
y′ + 2y = e−t, y(0) = 0
Example 10
Introduction IVP
EX 1.
EX 2.
IVP
Laplace Transform EX 3.
EX 4.
Table
Prop 1: Linearity EX 5.
EX 6.
Prop 2: Derivative EX 7.
EX 8.
EX 9.
EX 10.
Lecture 1 Sujin Khomrutai – 23 / 23
EX. Solve the IVP
y′′ + y = 3t, y(0) = 1, y′(0) = 2.
