Chapter 4: Q7E (page 157)

Question: Find a synchronous solution of the form $AcosΩt + BsinΩt$ to the given forced oscillator equation using the method of Example to solve forAand B $y'' + 2 y' + 4 y = 5 \sin 3 t, Ω = 3$ .
.

Short Answer

Expert verified

Answer

The synchronous solution is for the oscillator equation .

Step by step solution

Finding the differential equation of Y

Given differential equation isand $y'' + 2 y' + 4 y = 5 \sin 3 t$

The synchronous solution of the form $y = Acos 3 t + Bsin 3 t .$

Substitute the value of Y in the differential equation.

$y' (t) = - 3 Asin 3 t + 3 Bcos 3 t \dots (1) y'' (t) = - 9 Acos 3 t - 9 Bsin 3 t \dots (2)$

Substitute the y ' and y'' in the given equation.

Substitute and in the differential equation;

$(- 3 Asin 3 t + 3 Bcos 3 t) + 4 (Acos 3 t + Bsin 3 t) = 5 \sin 3 t - 5 A + 6 B) \cos 3 t + (- 6 A - 5 B) \sin 3 t = 5 \sin 3 t$

Equate the coefficients ofand;

$- 5 A + 6 B = 0 \dots (3) - 6 A - 5 B = 5 \dots (4)$

Finding the value of A

Multiply (3) with (4) and (5) with and then add two equations;

$- 25 A + 30 B = 0 - 36 A - 30 B = 30 61 A = 30 A = - \frac{30}{61}$

Finding the value of B

Substitute A in (3)

$- 5 (- \frac{30}{61}) + 6 B = 0 \frac{150}{61} + 6 B = 0 6 B = - \frac{150}{61} B = - \frac{25}{61}$

Therefore, the solution 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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Question: Find a synchronous solution of the form $AcosΩt + BsinΩt$ to the given forced oscillator equation using the method of Example to solve forAand B $y'' + 2 y' + 4 y = 5 \sin 3 t, Ω = 3$ .
.

Short Answer

Step by step solution

Finding the differential equation of Y

Substitute the y ' and y'' in the given equation.

Finding the value of A

Finding the value of B

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

Statistics

Calculus

Theoretical and Mathematical Physics

Decision Maths

Study anywhere. Anytime. Across all devices.

Question: Find a synchronous solution of the form AcosΩt+BsinΩt to the given forced oscillator equation using the method of Example to solve forAand B y''+2y'+4y=5sin3t,Ω=3..

Short Answer

Step by step solution

Finding the differential equation of Y

Substitute the y ' and y'' in the given equation.

Finding the value of A

Finding the value of B

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

Statistics

Calculus

Theoretical and Mathematical Physics

Decision Maths

Study anywhere. Anytime. Across all devices.

Question: Find a synchronous solution of the form $AcosΩt + BsinΩt$ to the given forced oscillator equation using the method of Example to solve forAand B $y'' + 2 y' + 4 y = 5 \sin 3 t, Ω = 3$ .
.