Q11E Find a general solution for the ... [FREE SOLUTION]

91影视

Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 6: Q11E (page 332)

Find a general solution for the differential equation with x as the independent variable:
$y^{(4)} + 4 y''' + 6 y'' + 4 y' + y = 0$

Short Answer

Expert verified

The general solution for the differential equation with x as the independent variable is $y = C_{1} e^{鈭� x} + C_{2} x e^{鈭� x} + C_{3} x^{2} e^{鈭� x} + C_{4} x^{3} e^{鈭� x}$

Step by step solution

Auxiliary equation:

The given differential equation is $y^{(4)} + 4 y''' + 6 y'' + 4 y' + y = 0$ . To solve this equation, we look at its auxillary equation which is $m^{4} + 4 m^{3} + 6 m^{2} + 4 m + 1 = 0$ .

By binomial theorem, it is clear seen that the auxillary equation is equal to ${(m + 1)}^{4}$ . So, $m = 鈭� 1, 鈭� 1, 鈭� 1, 鈭� 1$ . In other words, -1 is a multiple root repeasted four times.

General solution:

The general solution to the given differential equation is given by

$y = C_{1} e^{鈭� x} + C_{2} x e^{鈭� x} + C_{3} x^{2} e^{鈭� x} + C_{4} x^{3} e^{鈭� x}$ , where $C_{i} (1 鈮� i 鈮� 4)$ are arbitrary constant.

The general solution of the given differential equation is $y = C_{1} e^{鈭� x} + C_{2} x e^{鈭� x} + C_{3} x^{2} e^{鈭� x} + C_{4} x^{3} e^{鈭� x}$

Hence the final solution is $y = C_{1} e^{鈭� x} + C_{2} x e^{鈭� x} + C_{3} x^{2} e^{鈭� x} + C_{4} x^{3} e^{鈭� x}$

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find a general solution for the differential equation with x as the independent variable:
$y^{(4)} + 4 y''' + 6 y'' + 4 y' + y = 0$

Short Answer

Step by step solution

Auxiliary equation:

General solution:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Logic and Functions

Mechanics Maths

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

91影视

Find a general solution for the differential equation with x as the independent variable:y(4)+4y'''+6y''+4y'+y=0

Short Answer

Step by step solution

Auxiliary equation:

General solution:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Logic and Functions

Mechanics Maths

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

Find a general solution for the differential equation with x as the independent variable:
$y^{(4)} + 4 y''' + 6 y'' + 4 y' + y = 0$