Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 4

For the case that a function has multiple roots, \(f\left(x_{i}\right)=0, i=1,2, \ldots\), it can be shown that $$ \delta(f(x))=\sum_{i=1}^{n} \frac{\delta\left(x-x_{i}\right)}{\left|f^{\prime}\left(x_{i}\right)\right|} $$ Use this result to evaluate \(\int_{-\infty}^{\infty} \delta\left(x^{2}-5 x-6\right)\left(3 x^{2}-7 x+2\right) d x\).

Short Answer

Expert verified
The integral evaluates to \(8/3\)

Step by step solution

01

Find the roots of the function inside the delta function

The function inside the delta function is \(x^2-5x-6\). To find the roots of this function, it needs to be equated to zero and solved for \(x\). This gives \(x^2-5x-6=0\). The roots of this equation can be found using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here \(a=1\), \(b=-5\) and \(c=-6\). Plugging these values into the quadratic formula and simplifying gives the roots \(x_1=6\), \(x_2=-1\). These are the points where \(x^2-5x-6\) is zero, so they can be used to express the delta function in the integral.
02

Find the derivative of the function inside the delta function

The derivative of the function is needed to evaluate the expression for the delta function. The derivative of \(x^2-5x-6\) with respect to \(x\) is \(2x-5\). This function needs to be evaluated at the roots found in step 1. This gives \(f'(x_1)=7\) and \(f'(x_2)=3\). These values can be substituted into the expression for the delta function.
03

Write the integral using the delta function property and solve

The given integral can now be written using the delta function property as \( \int_{- \infty}^{\infty} \left(\frac{\delta(x - 6)}{|7|} + \frac{\delta(x + 1)}{|3|}\right) (3x^2 - 7x + 2) dx\). Using the property of the delta function, \(\int_{- \infty}^{\infty} \delta(x - a)f(x) dx = f(a)\), this integral can be solved by simply substituting \(x\) with the argument of the delta function and ignoring the absolute value signs. This gives the solution \((3*6^2 - 7*6 + 2)/7 + (3*(-1)^2 - 7*(-1) + 2) / 3 = 4 - 4/3 = 8/3 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For \(a>0\), find the Fourier transform, \(\hat{f}(k)\), of \(f(x)=e^{-a|x|}\).

For the following problems, draw the given function and find the Laplace transform in closed form. a. \(f(t)=1+\sum_{n=1}^{\infty}(-1)^{n} H(t-n)\). b. \(f(t)=\sum_{n=0}^{\infty}[H(t-2 n+1)-H(t-2 n)] .\) $$ \begin{aligned} &\text { c. } f(t)=\sum_{n=0}^{\infty}(t-2 n)[H(t-2 n)-H(t-2 n-1)]+(2 n+2-t)[H(t-1 \\ &2 n-1)-H(t-2 n-2)] \end{aligned} $$

Show that the convolution operation is associative: \((f *(g * h))(t)=\) \(((f * g) * h)(t)\)

Use Laplace transforms to convert the following system of differential equations into an algebraic system, and find the solution of the differential equations. $$ \begin{array}{ll} x^{\prime \prime}=3 x-6 y, & x(0)=1, & x^{\prime}(0)=0 \\ y^{\prime \prime}=x+y, & y(0)=0, & y^{\prime}(0)=0 \end{array} $$

Use the result from Problem 6 plus properties of the Fourier transform to find the Fourier transform, of \(f(x)=x^{2} e^{-a|x|}\) for \(a>0\).
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free