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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: Q8E (page 157)

Question: Find a synchronous solution of the form $础肠辞蝉惟迟 + 叠蝉颈苍惟迟$ to the given forced oscillator equation using the method of Example to solve forAand B $y'' + 2 y' + 5 y = - 50 \sin 5 t, 惟 = 5$ .
.

Short Answer

Expert verified

Answer

The synchronous solution to the given forced oscillator is;

$y = \cos 5 t - 2 \sin 5 t$

Step by step solution

01

Step 1: Finding the differential equation of Y

Given differential equation is $y'' + 2 y' + 5 y = - 50 \sin 5 t$ and

The synchronous solution of the formdata-custom-editor="chemistry" $y = Acos 5 t + Bsin 5 t .$

Substitute the value ofin the differential equation,

$y' (t) = - 5 Asin 5 t + 5 Bcos 5 t 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� (1) y'' (t) = - 25 Acos 5 t - 25 Bsin 5 t 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� (2)$

02

Substitute the and in the given equation

$- 25 Acos 5 t - 25 Bsin 5 t + 2 (- 5 Asin 5 t + 5 Bcos 5 t) + 5 (Acos 5 t + Bsin 5 t) = - 50 \sin 5 t$

Equate the coefficients ofand

03

Step 3: Finding the value of

Multiplywith and add two equations,

$- 40 A + 20 B = 0 - 10 A - 20 B = - 50 - 50 A = - 50 A = 1$

04

Step 4: Finding the value of B

Substitute in

$- 20 (1) + 10 B = 0 10 B = 20 B = 2$

Therefore, the solution is $y = \cos 5 t - 2 \sin 5 t$ .

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