Chapter 4: Q9E (page 157)

Question: Find a synchronous solution of the form $AcosΩt + BsinΩt$ to the given forced oscillator equation using the method of Example 4 to solve for A and B $y'' + 2 y' + 4 y = 6 \cos 2 t + 8 \sin 2 t, Ω = 2$ .
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Short Answer

Expert verified

Answer

The synchronous solution to the given forced oscillator equation is $y (t) = - 2 \cos 2 t + \frac{3}{2} \sin 2 t .$

Step by step solution

Finding the differential equation of Y

Given differential equation is $y'' + 2 y' + 4 y = 6 \cos 2 t + 8 \sin 2 t, Ω = 2 .$

The synchronous solution of the form $y = Acos 2 t + Bsin 2 t .$

Now substitute the value of y in the differential equation;

$y' (t) = - 2 Asin 2 t + 2 Bcos 2 t \dots (1) y'' (t) = - 4 Acos 2 t - 4 Bsin 2 t \dots (2)$

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

Substitute and in the differential equation;

$- 4 Acos 2 t - 4 Bsin 2 t + 2 (- 2 Asin 2 t + 2 Bcos 2 t) + 4 (Acos 2 t + Bsin 2 t) = 6 \cos 2 t + 8 \sin 2 t 4 Bcos 2 t - 4 Asin 2 t = 6 \cos 2 t + 8 \sin 2 t$

Step 3: Finding the value of B

Equate the coefficients of $\cos 2 t$ , then $4 B = 6$ we get

$B = \frac{6}{4} = \frac{3}{2}$

Step 4: Finding the value of A

Equate the coefficients of sin2t , then we get;

$- 4 A = 8 A = - \frac{8}{4} A = - 2$

Therefore, the solution is $y (t) = - 2 \cos 2 t + \frac{3}{2} \sin 2 t$ .

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Question: Find a synchronous solution of the form $AcosΩt + BsinΩt$ to the given forced oscillator equation using the method of Example 4 to solve for A and B $y'' + 2 y' + 4 y = 6 \cos 2 t + 8 \sin 2 t, Ω = 2$ .
.

Short Answer

Step by step solution

Finding the differential equation of Y

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

Step 3: Finding the value of B

Step 4: Finding the value of A

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Question: Find a synchronous solution of the form AcosΩt+BsinΩtto the given forced oscillator equation using the method of Example 4 to solve for A and By''+2y'+4y=6cos2t+8sin2t,Ω=2..

Short Answer

Step by step solution

Finding the differential equation of Y

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

Step 3: Finding the value of B

Step 4: Finding the value of A

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Mechanics Maths

Logic and Functions

Discrete Mathematics

Geometry

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 4 to solve for A and B $y'' + 2 y' + 4 y = 6 \cos 2 t + 8 \sin 2 t, Ω = 2$ .
.