Handbook of Integral Equations

Oct 13, 2025 by windailys.com 57 views

Introduction

In this article, we will focus on solving the initial value problem (IVP) of a second-order linear differential equation with a Dirac delta function. The given differential equation is:

\frac{d^2 y}{d t^2} + 64y = \delta(t - k\pi), \quad y(0) = 0, \quad y'(0) = 6

where $\delta(t - k\pi)$ is the Dirac delta function. The Laplace transform of the solution is given as $Y(s) = \square$ . Our goal is to find the solution $y(t)$ and the Laplace transform $Y(s)$ .

The Dirac Delta Function

The Dirac delta function $\delta(t - k\pi)$ is a generalized function that is zero everywhere except at $t = k\pi$ , where it is infinite. The Dirac delta function has the following properties:

$\delta(t - k\pi) = 0$ for $t \neq k\pi$
$\int_{-\infty}^{\infty} \delta(t - k\pi) dt = 1$
$\int_{-\infty}^{\infty} f(t) \delta(t - k\pi) dt = f(k\pi)$

The Laplace Transform

The Laplace transform of a function $f(t)$ is defined as:

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

The Laplace transform has the following properties:

$F(s) = \int_{0}^{\infty} f(t) e^{-st} dt$
$F(s) = \mathcal{L}\{f(t)\}$
$\mathcal{L}\{e^{at}\} = \frac{1}{s - a}$

Solving the IVP

To solve the IVP, we will use the Laplace transform method. We will first take the Laplace transform of the differential equation and then solve for $Y(s)$ .

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

\mathcal{L}\{\frac{d^2 y}{d t^2} + 64y\} = \mathcal{L}\{\delta(t - k\pi)\}

Using the linearity property of the Laplace transform, we can write:

\mathcal{L}\{\frac{d^2 y}{d t^2}\} + 64\mathcal{L}\{y\} = \mathcal{L}\{\delta(t - k\pi)\}

Using the property of the Laplace transform of the second derivative, we can write:

s^2 Y(s) - sy(0) - y'(0) + 64Y(s) = e^{-ks\pi}

Substituting the initial conditions, we get:

s^2 Y(s) - 6 + 64Y(s) = e^{-ks\pi}

Solving for $Y(s)$ , we get:

Y(s) = \frac{6 + e^{-ks\pi}}{s^2 + 64}

The Solution

To find the solution $y(t)$ , we will take the inverse Laplace transform of $Y(s)$ .

Using the property of the inverse Laplace transform, we can write:

y(t) = \mathcal{L}^{-1}\{Y(s)\}

Substituting the expression for $Y(s)$ , we get:

y(t) = \mathcal{L}^{-1}\{\frac{6 + e^{-ks\pi}}{s^2 + 64}\}

Using the property of the inverse Laplace transform of the rational function, we can write:

y(t) = 6\mathcal{L}^{-1}\{\frac{1}{s^2 + 64}\} + \mathcal{L}^{-1}\{\frac{e^{-ks\pi}}{s^2 + 64}\}

Using the property of the inverse Laplace transform of the exponential function, we can write:

y(t) = 6\mathcal{L}^{-1}\{\frac{1}{s^2 + 64}\} + e^{-k\pi t}\mathcal{L}^{-1}\{\frac{1}{s^2 + 64}\}

Using the property of the inverse Laplace transform of the rational function, we can write:

y(t) = 6\frac{\sin(8t)}{8} + e^{-k\pi t}\frac{\sin(8t)}{8}

Conclusion

In this article, we have solved the initial value problem (IVP) of a second-order linear differential equation with a Dirac delta function. We have used the Laplace transform method to solve for the solution $y(t)$ and the Laplace transform $Y(s)$ . The solution is given by:

y(t) = 6\frac{\sin(8t)}{8} + e^{-k\pi t}\frac{\sin(8t)}{8}

The Laplace transform is given by:

Y(s) = \frac{6 + e^{-ks\pi}}{s^2 + 64}

This solution can be used to model various physical systems, such as electrical circuits and mechanical systems, where a Dirac delta function is used to represent a sudden change in the system.

References

[1] A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, 2008.
[2] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 2007.
[3] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, 2007.

Q: What is the Dirac delta function and how is it used in the IVP?

A: The Dirac delta function $\delta(t - k\pi)$ is a generalized function that is zero everywhere except at $t = k\pi$ , where it is infinite. It is used in the IVP to represent a sudden change in the system.

Q: What is the Laplace transform and how is it used to solve the IVP?

A: The Laplace transform is a mathematical tool used to solve differential equations. It is defined as:

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

The Laplace transform is used to solve the IVP by taking the Laplace transform of the differential equation and then solving for $Y(s)$ .

Q: How do you take the Laplace transform of the second derivative?

A: To take the Laplace transform of the second derivative, you can use the property of the Laplace transform of the second derivative:

\mathcal{L}\{\frac{d^2 y}{d t^2}\} = s^2 Y(s) - sy(0) - y'(0)

Q: How do you solve for $Y(s)$ in the IVP?

A: To solve for $Y(s)$ in the IVP, you can use the property of the Laplace transform of the rational function:

Y(s) = \frac{6 + e^{-ks\pi}}{s^2 + 64}

Q: How do you take the inverse Laplace transform of $Y(s)$ to find the solution $y(t)$ ?

A: To take the inverse Laplace transform of $Y(s)$ , you can use the property of the inverse Laplace transform of the rational function:

y(t) = 6\mathcal{L}^{-1}\{\frac{1}{s^2 + 64}\} + e^{-k\pi t}\mathcal{L}^{-1}\{\frac{1}{s^2 + 64}\}

Q: What is the solution $y(t)$ and how is it obtained?

A: The solution $y(t)$ is obtained by taking the inverse Laplace transform of $Y(s)$ :

y(t) = 6\frac{\sin(8t)}{8} + e^{-k\pi t}\frac{\sin(8t)}{8}

Q: What is the Laplace transform $Y(s)$ and how is it obtained?

A: The Laplace transform $Y(s)$ is obtained by solving for $Y(s)$ in the IVP:

Y(s) = \frac{6 + e^{-ks\pi}}{s^2 + 64}

Q: What are some common applications of the IVP with a Dirac delta function?

A: The IVP with a Dirac delta function has many applications in physics and engineering, such as:

Modeling electrical circuits with sudden changes in voltage or current
Modeling mechanical systems with sudden changes in force or displacement
Modeling control systems with sudden changes in input or output

Q: What are some common challenges in solving the IVP with a Dirac delta function?

A: Some common challenges in solving the IVP with a Dirac delta function include:

Handling the singularity of the Dirac delta function
Dealing with the sudden change in the system
Ensuring that the solution satisfies the initial conditions

Q: What are some common tools and techniques used to solve the IVP with a Dirac delta function?

A: Some common tools and techniques used to solve the IVP with a Dirac delta function include:

Laplace transform method
Inverse Laplace transform method
Rational function method
Exponential function method

Q: What are some common mistakes to avoid when solving the IVP with a Dirac delta function?

A: Some common mistakes to avoid when solving the IVP with a Dirac delta function include:

Failing to handle the singularity of the Dirac delta function
Ignoring the sudden change in the system
Failing to ensure that the solution satisfies the initial conditions

Q: What are some common resources for learning more about solving the IVP with a Dirac delta function?

A: Some common resources for learning more about solving the IVP with a Dirac delta function include:

Textbooks on differential equations and Laplace transforms
Online tutorials and videos
Research papers and articles
Online forums and communities