Chapter 8: Q5.23P (page 415)

Find the general solutions of the following equations and compare computer solutions.
$(D^{2} + 1) (D^{2} - 1) y = 0$ Hint: $D^{2} + 1 = (D + i) (D - i)$ .

Short Answer

Expert verified

The general solution is $y = c_{1} e^{- i x} + c_{2} e^{i x} + c_{3} e^{x} + c_{4} e^{x}$ .

Step by step solution

Given information from question

Given equation is $(D^{2} + 1) (D^{2} - 1) y = 0$ .

Differential equation

A differential equation is a formula that connects the derivatives of one or more unknown functions. Functions are used to represent physical quantities, derivatives are used to characterise their rates of change, and differential equations are used to define a relationship between them in applications.

Calculate the general solution

First expand the differential equation

$(D + i) (D - i) (D + 1) (D - 1) y = 0$

It is found that the general solution for any differential that has an auxiliary equation with unequal roots is

$y = c_{1} e^{a x} + c_{2} e^{b x} + \dots . . + c_{n} e^{a_{n} x}$

In our cases it is $a_{1} = - i, a_{2} = i, a_{3} = - 1, a_{2} = 1$ .Then, the general solution is,

$y = c_{1} e^{- i x} + c_{2} e^{i x} + c_{3} e^{x} + c_{4} e^{x}$ .

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!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Find the general solutions of the following equations and compare computer solutions.
$(D^{2} + 1) (D^{2} - 1) y = 0$ Hint: $D^{2} + 1 = (D + i) (D - i)$ .

Short Answer

Step by step solution

Given information from question

Differential equation

Calculate the general solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Engineering Physics

Quantum Physics

Medical Physics

Nuclear Physics

Measurements

Space Physics

Study anywhere. Anytime. Across all devices.

Find the general solutions of the following equations and compare computer solutions.D2+1D2-1y=0Hint:D2+1=(D+i)(D-i).

Short Answer

Step by step solution

Given information from question

Differential equation

Calculate the general solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Engineering Physics

Quantum Physics

Medical Physics

Nuclear Physics

Measurements

Space Physics

Study anywhere. Anytime. Across all devices.

Find the general solutions of the following equations and compare computer solutions.
$(D^{2} + 1) (D^{2} - 1) y = 0$ Hint: $D^{2} + 1 = (D + i) (D - i)$ .