Q17E Find a general solution to the g... [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: Q17E (page 332)

Find a general solution to the givenhomogeneous equation. $(D + 4) (D 鈭� 3) {(D + 2)}^{3} {(D^{2} + 4 D + 5)}^{2} D^{5} [y] = 0$

Short Answer

Expert verified

The answer to this problem is:

$y = c_{1} e^{鈭� 4 x} + c_{2} e^{3 x} + (c_{3} + c_{4} x + c_{5} x^{2}) e^{鈭� 2 x} + (c_{6} + c_{7} x) e^{鈭� 2 x} \cos x + (c_{8} + c_{9} x) \sin x + c_{10} + c_{11} x + c_{12} x^{2} + c_{13} x^{3} + c_{14} x^{4}$

Step by step solution

Homogenous Equation

A homogeneous system of linear equations is one in which all of the constant terms are zero. A homogeneous system always has at least one solution, namely the zero vector. When a row operation is applied to a homogeneous system, the new system is still homogeneous.

Solving of Homogenous Equation:

We will have to solve the following homogenous equation here;

$\begin{array}{l} D + 4) (D - 3) {(D + 2)}^{2} {(D^{2} + 4 D + 5)}^{2} D^{5} [y] = 0 \\ (r + 4) (r 鈭� 3) {(r + 2)}^{3} {(r^{2} + 4 r + 5)}^{2} r^{5} = 0 \\ r_{1} = - 4 \\ r_{2} = 3 \\ r_{3, 4, 5} = 鈭� 2 \\ r^{2} + 4 r + 5 = 0 \\ r_{6, 7, 8, 9} = \frac{鈭� 4 卤 \sqrt{4^{2} 鈭� 4.5}}{2} \\ = 鈭� 2 卤 i \\ r_{10, 11, 12, 13, 14} = 0 \\ y = c_{1} e^{鈭� 4 x} + c_{2} e^{3 x} + (c_{3} + c_{4} x + c_{5} x^{2}) e^{鈭� 2 x} + (c_{6} + c_{7} x) e^{鈭� 2 x} \cos x + (c_{8} + c_{9} x) \sin x + c_{10} + c_{11} x + c_{12} x^{2} + c_{13} x^{3} + c_{14} x^{4} \end{array}$

Hence, the final answer is:

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 to the givenhomogeneous equation. $(D + 4) (D 鈭� 3) {(D + 2)}^{3} {(D^{2} + 4 D + 5)}^{2} D^{5} [y] = 0$

Short Answer

Step by step solution

Homogenous Equation

Solving of Homogenous Equation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Pure Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

91影视

Find a general solution to the givenhomogeneous equation.(D+4)(D鈭�3)(D+2)3(D2+4D+5)2D5[y]=0

Short Answer

Step by step solution

Homogenous Equation

Solving of Homogenous Equation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Pure Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Find a general solution to the givenhomogeneous equation. $(D + 4) (D 鈭� 3) {(D + 2)}^{3} {(D^{2} + 4 D + 5)}^{2} D^{5} [y] = 0$