Question

Question: Find a synchronous solution of the form $\mathbf{Acos\Omega t}\mathbf{+}\mathbf{Bsin\Omega t}$to the given forced oscillator equation using the method of Example to solve forAand B$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathbf{y}\mathbf{=}\mathbf{-}\mathbf{50}\mathbf{sin}\mathbf{5}\mathbf{t}\mathbf{,}\mathbf{\Omega }\mathbf{=}\mathbf{5}$ .

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Short answer

Answer

The synchronous solution to the given forced oscillator is;

$\mathbf{y}\mathbf{=}\mathbf{cos}\mathbf{5}\mathbf{t}\mathbf{-}\mathbf{2}\mathbf{sin}\mathbf{5}\mathbf{t}$

Step-by-step solution

Step 1: Finding the differential equation of  Y

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

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

Substitute the value ofin the differential equation,

$$\begin{gathered} \mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{5}\mathbf{Asin}\mathbf{5}\mathbf{t}\mathbf{+}\mathbf{5}\mathbf{Bcos}\mathbf{5}\mathbf{t} \mathbf{\dots }\mathbf{\left(}\mathbf{1}\mathbf{\right)} \\ \mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{25}\mathbf{Acos}\mathbf{5}\mathbf{t}\mathbf{-}\mathbf{25}\mathbf{Bsin}\mathbf{5}\mathbf{t} \mathbf{\dots }\mathbf{\left(}\mathbf{2}\mathbf{\right)} \end{gathered}$$

Substitute the  and   in the given equation

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

Equate the coefficients ofand

Step 3: Finding the value of 

Multiplywith and add two equations,

$$\begin{gathered} \mathbf{-}\mathbf{40}\mathbf{A}\mathbf{+}\mathbf{20}\mathbf{B}\mathbf{=}\mathbf{0} \\ \mathbf{-}\mathbf{10}\mathbf{A}\mathbf{-}\mathbf{20}\mathbf{B}\mathbf{=}\mathbf{-}\mathbf{50} \\ \mathbf{-}\mathbf{50}\mathbf{A}\mathbf{=}\mathbf{-}\mathbf{50} \\ \mathbf{A}\mathbf{=}\mathbf{1} \end{gathered}$$

 Step 4: Finding the value of  B

Substitute in

$$\begin{gathered} \mathbf{-}\mathbf{20}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{+}\mathbf{10}\mathbf{B}\mathbf{=}\mathbf{0} \\ \mathbf{10}\mathbf{B}\mathbf{=}\mathbf{20} \\ \mathbf{B}\mathbf{=}\mathbf{2} \end{gathered}$$

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