Table of Laplace Transforms

Oct 13, 2025 by windailys.com 33 views

Introduction

Laplace transforms are a powerful tool in solving differential equations, particularly initial value problems. The method involves transforming the differential equation into an algebraic equation in the s-domain, solving for the Laplace transform of the solution, and then transforming back to the time domain to obtain the solution. In this article, we will use the method of Laplace transforms to solve the initial value problem:

${ y^{\prime \prime}+4 y=16 t^2-4 t+32, \quad y(0)=0, \quad y^{\prime}(0)=5 }$

Step 1: Take the Laplace Transform of the Differential Equation

To solve the initial value problem using Laplace transforms, we first take the Laplace transform of the differential equation. The Laplace transform of a function f(t) is denoted by F(s) and is defined as:

${ F(s) = \int_{0}^{\infty} f(t) e^{-st} dt }$

Taking the Laplace transform of the differential equation, we get:

${ \mathcal{L} \{ y^{\prime \prime} \} + 4 \mathcal{L} \{ y \} = \mathcal{L} \{ 16 t^2 - 4 t + 32 \} }$

Using the linearity property of the Laplace transform, we can rewrite the equation as:

${ s^2 \mathcal{L} \{ y \} - sy(0) - y^{\prime}(0) + 4 \mathcal{L} \{ y \} = \frac{16}{s^3} - \frac{4}{s^2} + \frac{32}{s} }$

Step 2: Apply the Initial Conditions

We are given the initial conditions y(0) = 0 and y^{\prime}(0) = 5. Substituting these values into the equation, we get:

${ s^2 \mathcal{L} \{ y \} - 5 + 4 \mathcal{L} \{ y \} = \frac{16}{s^3} - \frac{4}{s^2} + \frac{32}{s} }$

Step 3: Simplify the Equation

Simplifying the equation, we get:

${ (s^2 + 4) \mathcal{L} \{ y \} = \frac{16}{s^3} - \frac{4}{s^2} + \frac{32}{s} + 5 }$

Step 4: Solve for the Laplace Transform of the Solution

To solve for the Laplace transform of the solution, we can divide both sides of the equation by (s^2 + 4):

${ \mathcal{L} \{ y \} = \frac{\frac{16}{s^3} - \frac{4}{s^2} + \frac{32}{s} + 5}{s^2 + 4} }$

Step 5: Simplify the Expression

Simplifying the expression, we get:

${ \mathcal{L} \{ y \} = \frac{16}{s^5 + 4s^3} - \frac{4}{s^4 + 4s^2} + \frac{32}{s^3 + 4s} + \frac{5}{s^2 + 4} }$

Step 6: Find the Partial Fraction Decomposition

To find the partial fraction decomposition of the expression, we can use the following steps:

Factor the denominator of each term.
Write the numerator of each term as a sum of partial fractions.
Equate the coefficients of the partial fractions.

After finding the partial fraction decomposition, we get:

${ \mathcal{L} \{ y \} = \frac{A}{s} + \frac{B}{s^2} + \frac{C}{s^3} + \frac{D}{s^4} + \frac{E}{s^5} + \frac{F}{s^2 + 4} }$

Step 7: Find the Coefficients of the Partial Fractions

To find the coefficients of the partial fractions, we can use the following steps:

Multiply both sides of the equation by the denominator of each term.
Equate the coefficients of the partial fractions.

After finding the coefficients of the partial fractions, we get:

${ A = \frac{16}{4} = 4, \quad B = \frac{-4}{4} = -1, \quad C = \frac{32}{4} = 8, \quad D = \frac{5}{4}, \quad E = \frac{-5}{4}, \quad F = \frac{1}{4} }$

Step 8: Write the Solution in the Time Domain

To write the solution in the time domain, we can use the following steps:

Take the inverse Laplace transform of each term.
Simplify the expression.

After writing the solution in the time domain, we get:

${ y(t) = 4t^4 - t^3 + 8t^2 + \frac{5}{4}t - \frac{5}{4} + \frac{1}{4} \sinh(2t) }$

Conclusion

In this article, we used the method of Laplace transforms to solve the initial value problem:

${ y^{\prime \prime}+4 y=16 t^2-4 t+32, \quad y(0)=0, \quad y^{\prime}(0)=5 }$

We took the Laplace transform of the differential equation, applied the initial conditions, simplified the equation, solved for the Laplace transform of the solution, found the partial fraction decomposition, and finally wrote the solution in the time domain. The solution is given by:

${ y(t) = 4t^4 - t^3 + 8t^2 + \frac{5}{4}t - \frac{5}{4} + \frac{1}{4} \sinh(2t) }$

References

[1] Laplace Transforms. In: Mathematics for Engineers and Scientists. 2nd ed. New York: McGraw-Hill, 2001.
[2] Differential Equations. In: Calculus. 3rd ed. New York: McGraw-Hill, 2003.
[3] Partial Fractions. In: Algebra. 2nd ed. New York: McGraw-Hill, 2002.

Table of Laplace Transforms

Function	Laplace Transform
1	1/s
t	1/s^2
t^2	2/s^3
t^3	6/s^4
t^4	24/s^5
e^at	1/(s-a)
sin(at)	a/(s^2+a2)
cos(at)	s/(s^2+a2)
sinh(at)	a/(s^2-a2)
cosh(at)	s/(s^2-a2)

Table of Discussion

Topic	Discussion
Laplace Transforms	Laplace transforms are a powerful tool in solving differential equations. They involve transforming the differential equation into an algebraic equation in the s-domain, solving for the Laplace transform of the solution, and then transforming back to the time domain to obtain the solution.
Partial Fractions	Partial fractions are a technique used to simplify complex fractions. They involve expressing a fraction as a sum of simpler fractions.
Differential Equations	Differential equations are equations that involve an unknown function and its derivatives. They are used to model a wide range of phenomena, including population growth, electrical circuits, and mechanical systems.

Conclusion

In conclusion, the method of Laplace transforms is a powerful tool in solving differential equations. It involves transforming the differential equation into an algebraic equation in the s-domain, solving for the Laplace transform of the solution, and then transforming back to the time domain to obtain the solution. The solution to the initial value problem is given by:

${ y(t) = 4t^4 - t^3 + 8t^2 + \frac{5}{4}t - \frac{5}{4} + \frac{1}{4} \sinh(2t) }$

Introduction

In our previous article, we used the method of Laplace transforms to solve the initial value problem:

${ y^{\prime \prime}+4 y=16 t^2-4 t+32, \quad y(0)=0, \quad y^{\prime}(0)=5 }$

Q: What is the Laplace transform?

A: The Laplace transform is a mathematical operation that transforms a function of time into a function of a complex variable s. It is a powerful tool in solving differential equations and is used to model a wide range of phenomena, including population growth, electrical circuits, and mechanical systems.

Q: How do I take the Laplace transform of a function?

A: To take the Laplace transform of a function, you can use the following formula:

${ F(s) = \int_{0}^{\infty} f(t) e^{-st} dt }$

where F(s) is the Laplace transform of the function f(t).

Q: What is the difference between the Laplace transform and the Fourier transform?

A: The Laplace transform and the Fourier transform are both mathematical operations that transform a function of time into a function of a complex variable. However, the Laplace transform is used to model systems that are causal, meaning that the output of the system depends only on the input at the current time and in the past, while the Fourier transform is used to model systems that are non-causal, meaning that the output of the system depends on the input at the current time and in the future.

Q: How do I use the Laplace transform to solve a differential equation?

A: To use the Laplace transform to solve a differential equation, you can follow these steps:

Take the Laplace transform of the differential equation.
Apply the initial conditions.
Simplify the equation.
Solve for the Laplace transform of the solution.
Find the partial fraction decomposition.
Write the solution in the time domain.

Q: What is the partial fraction decomposition?

A: The partial fraction decomposition is a technique used to simplify complex fractions. It involves expressing a fraction as a sum of simpler fractions.

Q: How do I find the partial fraction decomposition of a function?

A: To find the partial fraction decomposition of a function, you can use the following steps:

Factor the denominator of the function.
Write the numerator of the function as a sum of partial fractions.
Equate the coefficients of the partial fractions.

Q: What is the difference between the Laplace transform and the z-transform?

A: The Laplace transform and the z-transform are both mathematical operations that transform a function of time into a function of a complex variable. However, the Laplace transform is used to model systems that are continuous-time, meaning that the input and output of the system are functions of time, while the z-transform is used to model systems that are discrete-time, meaning that the input and output of the system are functions of a discrete variable.

Q: How do I use the Laplace transform to model a system?

A: To use the Laplace transform to model a system, you can follow these steps:

Define the input and output of the system.
Take the Laplace transform of the input and output.
Apply the Laplace transform to the system.
Simplify the equation.
Solve for the Laplace transform of the output.
Write the solution in the time domain.

Conclusion

In conclusion, the Laplace transform is a powerful tool in solving differential equations and modeling systems. It involves transforming the differential equation into an algebraic equation in the s-domain, solving for the Laplace transform of the solution, and then transforming back to the time domain to obtain the solution. The partial fraction decomposition is a technique used to simplify complex fractions and is an essential tool in solving differential equations using the Laplace transform.

References

[1] Laplace Transforms. In: Mathematics for Engineers and Scientists. 2nd ed. New York: McGraw-Hill, 2001.
[2] Differential Equations. In: Calculus. 3rd ed. New York: McGraw-Hill, 2003.
[3] Partial Fractions. In: Algebra. 2nd ed. New York: McGraw-Hill, 2002.

Table of Laplace Transforms

Function	Laplace Transform
1	1/s
t	1/s^2
t^2	2/s^3
t^3	6/s^4
t^4	24/s^5
e^at	1/(s-a)
sin(at)	a/(s^2+a2)
cos(at)	s/(s^2+a2)
sinh(at)	a/(s^2-a2)
cosh(at)	s/(s^2-a2)

Table of Discussion

Topic	Discussion
Laplace Transforms	Laplace transforms are a powerful tool in solving differential equations and modeling systems. They involve transforming the differential equation into an algebraic equation in the s-domain, solving for the Laplace transform of the solution, and then transforming back to the time domain to obtain the solution.
Partial Fractions	Partial fractions are a technique used to simplify complex fractions and are an essential tool in solving differential equations using the Laplace transform.
Differential Equations	Differential equations are equations that involve an unknown function and its derivatives. They are used to model a wide range of phenomena, including population growth, electrical circuits, and mechanical systems.