Question

3c. For each differential equation, find the Laplace transform of the solution y:

y'' + 3y' + 2y = 5 sin(πx), y(0) = y'(0) = 0.​​

Answer 1

`y(x)=(5)/(3\pi)-(5\pi)/(27+3\pi^2)e^(-3x)-(15)/(9\pi+\pi^3)cos(\pi x)-(5)/(9+\pi^2)sin(\pi x)`

Explanation:

`y''+3y'+2y=5sin(\pi x),\: y(0)=y'(0)=0\\\\\mathcal{L}\{y''\}+3\mathcal{L}\{y'\}+2\mathcal{L}\{y\}=\mathcal{L}\{5sin(\pi x)\}\\\\s^2Y(s)-sy(0)-y'(0)+3[sY(s)-y(0)]+2Y(s)=(5\pi)/(s^2+\pi^2)\\\\s^2Y(s)+3sY(s)=(5\pi)/(s^2+\pi^2)\\ \\(s^2+3s)Y(s)=(5\pi)/(s^2+\pi^2)\\ \\Y(s)=(5\pi)/((s^2+\pi^2)(s^2+3s))\\ \\Y(s)=(5\pi)/(s(s+3)(s^2+\pi^2))`

Perform the partial fraction decomposition

`(5 \pi)/(s \left(s + 3\right) \left(s^(2) + \pi^(2)\right))=(A)/(s)+(B)/(s + 3)+(C s + D)/(s^(2) + \pi^(2))\\5\pi=s \left(s + 3\right) \left(C s + D\right) + s \left(s^(2) + \pi^(2)\right) B + \left(s + 3\right) \left(s^(2) + \pi^(2)\right) A\\\\5 \pi=s^(3) A + s^(3) B + s^(3) C + 3 s^(2) A + 3 s^(2) C + s^(2) D + \pi^(2) s A + \pi^(2) s B + 3 s D + 3 \pi^(2) A\\\\5\pi=s^3(A+B+C)+s^2(3A+3C+D)+s(\pi^2A+\pi^2B+3D)+3\pi^2A`

Solve for each constant

`\begin{cases} A + B + C = 0\\3 A + 3 C + D = 0\\\pi^(2) A + \pi^(2) B + 3 D = 0\\3 \pi^(2) A = 5 \pi \end{cases}`

`3\pi^2A=5\pi\\A=(5)/(3\pi)`

`A+B+C=0\\3A+3B+3C=0\\(3A+3B+3C=0)-(3A+3C+D=0)\\3B-D=0\\3B=D`

`\pi^2A+\pi^2B+3D=0\\\pi^2((5)/(3\pi))+\pi^2B+3(3B)=0\\(5\pi)/(3)+\pi^2B+9B=0\\ B(\pi^2+9)=-(5\pi)/(3)\\B=-(5\pi)/(3(\pi^2+9))\\B=-(5\pi)/(3\pi^2+27)`

`3B=D\\3(-(5\pi)/(3\pi^2+27))=D\\-(5\pi)/(\pi^2+9)=D`

`A+B+C=0\\(5)/(3\pi)-(5\pi)/(3\pi^2+27)+C=0\\(15\pi^2+135)/(3\pi(3\pi^2+27))-(15\pi^2)/((3\pi)(3\pi^2+27))+C=0\\(135)/(9\pi^3+81\pi)+C=0\\(15)/(\pi^3+9\pi)+C=0\\C=-(15)/(\pi^3+9\pi)`

Take the inverse transform and solve for the IVP

`Y(s)=((5)/(3 \pi))/(s)+(- (5 \pi)/(27 + 3 \pi^(2)))/(s + 3)+(- (15 s)/(9 \pi + \pi^(3)) - (5 \pi)/(9 + \pi^(2)))/(s^(2) + \pi^(2))\\\\Y(s)=((5)/(3 \pi))/(s)-((5 \pi)/(27 + 3 \pi^(2)))/(s + 3)-( (15 s)/(9 \pi + \pi^(3)))/(s^(2) + \pi^(2))-((5\pi)/(9+\pi^2) )/(s^2+\pi^2)\\ \\y(x)=(5)/(3\pi)-(5\pi)/(27+3\pi^2)e^(-3x)-(15)/(9\pi+\pi^3)cos(\pi x)-(5)/(9+\pi^2)sin(\pi x)`