Q25E Find a particular solution to th... [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: Q25E (page 180)

Find a particular solution to the differential equation.
$y'' + 2 y' + 4 y = 111 e^{2 t} \cos 3 t$

Short Answer

Expert verified

The particular solution is $y_{p} = e^{2 t} (\cos 3 t + 6 \sin 3 t)$ .

Step by step solution

Firstly, write the auxiliary equation of the above differential equation.

The differential equation is:

$y'' + 2 y' + 4 y = 111 e^{2 t} \cos 3 t 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� 鈥� (1)$

Write the homogeneous differential equation of the equation (1),

$y'' + 2 y' + 4 y = 0$

The auxiliary equation for the above equation,

$m^{2} + 2 m + 4 = 0$

Now find the roots of the auxiliary equation

Solve the auxiliary equation,

$\begin{matrix} m^{2} + 2 m + 4 = 0 \\ m = \frac{- 2 卤 \sqrt{2^{2} - 4 (1) (4)}}{2 (1)} \\ m = \frac{- 2 卤 \sqrt{- 12}}{2} \\ m = - 1 卤 i \sqrt{3} \end{matrix}$

The roots of the auxiliary equation are,

$m_{1} = - 1 + i \sqrt{3}, 鈥夆赌夆赌� & 鈥夆赌夆赌� m_{2} = - 1 - i \sqrt{3}$

The complementary solution of the given equation is,

$y_{c} = e^{- x} (c_{1} \cos \sqrt{3} x + c_{2} \sin \sqrt{3} x)$

Use the method of undetermined coefficients to find a particular solution to the differential equation.

According to the method of undetermined coefficients, assume, the particular solution of equation (1),

$y_{p} = e^{2 t} (A_{0} \cos 3 t + B_{0} \sin 3 t) 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌� ...... (2)$

Now find the derivative of the above equation,

$\begin{matrix} y_{p}' = 2 e^{2 t} (A_{0} \cos 3 t + B_{0} \sin 3 t) + e^{2 t} (- 3 A_{0} \sin 3 t + 3 B_{0} \cos 3 t) \\ y_{p}'' = 4 e^{2 t} (A_{0} \cos 3 t + B_{0} \sin 3 t) + 2 e^{2 t} (- 3 A_{0} \sin 3 t + 3 B_{0} \cos 3 t) \\ 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌� + 2 e^{2 t} (- 3 A_{0} \sin 3 t + 3 B_{0} \cos 3 t) + e^{2 t} (- 9 A_{0} \cos 3 t - 9 B_{0} \sin 3 t) \\ y_{p}'' = e^{2 t} (- 5 A_{0} \cos 3 t - 5 B_{0} \sin 3 t) + e^{2 t} (- 12 A_{0} \sin 3 t + 12 B_{0} \cos 3 t) \end{matrix}$

From the equation (1), Substitute the value of $y_{p}'', 鈥夆赌� y_{p}'$ and $鈥� y_{p}$ in the equation (1),

$\begin{array}{l} 鈬� y_{p}'' + 2 y_{p}' + 4 y_{p} = 111 e^{2 t} \cos 3 t \\ 鈬� e^{2 t} (- 5 A_{0} \cos 3 t - 5 B_{0} \sin 3 t) + e^{2 t} (- 12 A_{0} \sin 3 t + 12 B_{0} \cos 3 t) \\ 鈥夆赌夆赌夆€夆赌夆赌� + 2 [2 e^{2 t} (A_{0} \cos 3 t + B_{0} \sin 3 t) + e^{2 t} (- 3 A_{0} \sin 3 t + 3 B_{0} \cos 3 t)] \\ 鈥夆赌夆赌夆€夆赌夆赌� + 4 e^{2 t} (A_{0} \cos 3 t + B_{0} \sin 3 t) = 111 e^{2 t} \cos 3 t \\ 鈬� e^{2 t} (3 A_{0} \cos 3 t + 3 B_{0} \sin 3 t) + e^{2 t} (- 18 A_{0} \sin 3 t + 18 B_{0} \cos 3 t) = 111 e^{2 t} \cos 3 t \\ 鈬� e^{2 t} \cos 3 t (3 A_{0} + 18 B_{0}) + e^{2 t} \sin 3 t (3 B_{0} - 18 A_{0}) = 111 e^{2 t} \cos 3 t \end{array}$

Final conclusion:

Comparing all coefficients of the above equation;

$\begin{matrix} 3 A_{0} + 18 B_{0} = 111 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆€� . ..... (3) \\ 3 B_{0} - 18 A_{0} = 0 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌� . ..... (4) \end{matrix}$

Solve the above equations,

$\begin{matrix} (3 A_{0} + 18 B_{0}) = 111 脳 6 \\ 18 A_{0} + 108 B_{0} = 666 \\ - 18 A_{0} + 3 B_{0} 鈥夆赌夆赌夆€夆赌� = 0 \\ \overset{炉}{111 B_{0} = 666 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€�} \\ 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌� B_{0} = 6 \end{matrix}$

Substitute the value of $B_{0}$ in the equation (3),

$\begin{matrix} 3 A_{0} + 18 (6) = 111 \\ 3 A_{0} = 3 \\ A_{0} = 1 \end{matrix}$

Therefore, the particular solution of equation (1),

$\begin{matrix} y_{p} = e^{2 t} (A_{0} \cos 3 t + B_{0} \sin 3 t) \\ y_{p} = e^{2 t} (\cos 3 t + 6 \sin 3 t) \end{matrix}$

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 particular solution to the differential equation.
$y'' + 2 y' + 4 y = 111 e^{2 t} \cos 3 t$

Short Answer

Step by step solution

Firstly, write the auxiliary equation of the above differential equation.

Now find the roots of the auxiliary equation

Use the method of undetermined coefficients to find a particular solution to the differential equation.

Final conclusion:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Probability and Statistics

Geometry

Mechanics Maths

Calculus

Study anywhere. Anytime. Across all devices.

91影视

Find a particular solution to the differential equation.y''+2y'+4y=111e2tcos3t

Short Answer

Step by step solution

Firstly, write the auxiliary equation of the above differential equation.

Now find the roots of the auxiliary equation

Use the method of undetermined coefficients to find a particular solution to the differential equation.

Final conclusion:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Probability and Statistics

Geometry

Mechanics Maths

Calculus

Study anywhere. Anytime. Across all devices.

Find a particular solution to the differential equation.
$y'' + 2 y' + 4 y = 111 e^{2 t} \cos 3 t$