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Chapter 4: Q7E (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''+2y'+4y=5sin3t,惟=3.

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Short Answer

Expert verified

Answer

The synchronous solution is for the oscillator equation .

Step by step solution

01

Finding the differential equation of  Y

Given differential equation isandy''+2y'+4y=5sin3t

The synchronous solution of the formy=Acos3t+Bsin3t.

Substitute the value of Y in the differential equation.

y'(t)=-3Asin3t+3Bcos3t鈥�(1)y''(t)=-9Acos3t-9Bsin3t鈥�(2)

02

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

Substitute and in the differential equation;

(-3Asin3t+3Bcos3t)+4(Acos3t+Bsin3t)=5sin3t-5A+6B)cos3t+(-6A-5B)sin3t=5sin3t

Equate the coefficients ofand;

-5A+6B=0鈥�(3)-6A-5B=5鈥�(4)

03

Finding the value of  A

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

-25A+30B=0-36A-30B=3061A=30A=-3061

04

Finding the value of  B

Substitute A in (3)

-5-3061+6B=015061+6B=06B=-15061B=-2561

Therefore, the solution is.

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