Q4.3-19E Find a general solution y'''+y''... [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 4: Q4.3-19E (page 172)

Find a general solution $y''' + y'' + 3 y' - 5 y = 0$

Short Answer

Expert verified

The general solution of the given equation $y''' + y'' + 3 y' - 5 y = 0$ is $y (t) = C_{1} e^{t} + C_{2} e^{- t} \cos (2 t) + C_{3} e^{- t} \sin (2 t)$ .

Step by step solution

Using rational root theorem.

First, one needs to find the auxiliary equation and solve it. One has $r^{3} + r^{2} + 3 r - 5 = 0$ .

You can use the rational root theorem.

The first divisor of role="math" localid="1654073691473" $5$ is role="math" localid="1654073695156" $1$ if will be one solution of the equation and $(r - 1)$ will be a factor.

Indeed, you have $1^{3} + 1^{2} + 3 - 5 = 0$

Now you can dividerole="math" localid="1654073800144" $r^{3} + r^{2} + 3 r - 5$ byrole="math" localid="1654073795414" $r - 1$ to get $r^{2} + 2 r + 5$ .

Finding factors and roots.

The equation can be factored as $(r - 1) (r^{2} + 2 r + 5) = 0 .$

Since $r^{2} + 2 r + 5 = 0$

$r = \frac{- 2 卤 \sqrt{2^{2} - 4 脳 1 脳 5}}{} r = - 1 卤 2 i$

The roots of the auxiliary equation are $r = 1, r = - 1 + 2 i$ and $r = - 1 - 2 i$

Thus, the general solution of the differential equation is: $y (t) = C_{1} e^{t} + C_{2} e^{- t} \cos (2 t) + C_{3} e^{- t} \sin (2 t)$

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 $y''' + y'' + 3 y' - 5 y = 0$

Short Answer

Step by step solution

Using rational root theorem.

Finding factors and roots.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Pure Maths

Calculus

Mechanics Maths

Study anywhere. Anytime. Across all devices.

91影视

Find a general solution y'''+y''+3y'-5y=0

Short Answer

Step by step solution

Using rational root theorem.

Finding factors and roots.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Pure Maths

Calculus

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Find a general solution $y''' + y'' + 3 y' - 5 y = 0$