Q8E - Fundamentals Of Differential Equations And Boundary Value Problems

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 for  A and B $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathbf{y}\mathbf{=}\mathbf{-}\mathbf{50}\mathbf{sin}\mathbf{5}\mathbf{t}\mathbf{,}\mathbf{\Omega }\mathbf{=}\mathbf{5}$ .

 .

Step-by-Step Solution

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 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 form   data-custom-editor="chemistry" $\mathbf{y}\mathbf{=}\mathbf{Acos}\mathbf{5}\mathbf{t}\mathbf{+}\mathbf{Bsin}\mathbf{5}\mathbf{t}.$

 

Substitute the value of   in the differential equation,

 

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

Step 2: 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 of   and 

 

 

Step 3: Finding the value of

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