Question

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

y'' − y = 5e^−4x + 2x, y(0) = y'(0) = 0.

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Answer 1

`y(x)=-2x-(11)/(6)e^(-x)+(3)/(2)e^x+(1)/(3)e^(-4x)`

Explanation:

`y''-y=5e^(-4x)+2x,\: y(0)=y'(0)=0\\\\\mathcal{L}\{y''\}-\mathcal{L}\{y\}=\mathcal{L}\{5e^(-4x)\}+\mathcal{L}\{2x\}\\\\s^2Y(s)-sy(0)-y'(0)-Y(s)=(5)/(s+4)+(2)/(s^2)\\ \\s^2Y(s)-Y(s)=(5)/(s+4)+(2)/(s^2)\\ \\(s^2-1)Y(s)=(5)/(s+4)+(2)/(s^2)\\ \\Y(s)=(5)/((s+4)(s^2-1))+(2)/(s^2(s^2-1))\\ \\Y(s)=(5s^2+2(s+4))/(s^2(s+4)(s^2-1))\\ \\Y(s)=(5s^2+2s+8)/(s^2(s-1)(s+1)(s+4))`

Perform the partial fraction decomposition

`(5 s^(2) + 2 s + 8)/(s^(2) \left(s - 1\right) \left(s + 1\right) \left(s + 4\right))=(A)/(s)+(B)/(s^(2))+(C)/(s + 1)+(D)/(s - 1)+(E)/(s + 4)\\\\5 s^(2) + 2 s + 8=s^(2) \left(s - 1\right) \left(s + 1\right) E + s^(2) \left(s - 1\right) \left(s + 4\right) C + s^(2) \left(s + 1\right) \left(s + 4\right) D + s \left(s - 1\right) \left(s + 1\right) \left(s + 4\right) A + \left(s - 1\right) \left(s + 1\right) \left(s + 4\right) B`

`5 s^(2) + 2 s + 8=s^(4) A + s^(4) C + s^(4) D + s^(4) E + 4 s^(3) A + s^(3) B + 3 s^(3) C + 5 s^(3) D - s^(2) A + 4 s^(2) B - 4 s^(2) C + 4 s^(2) D - s^(2) E - 4 s A - s B - 4 B\\\\5 s^(2) + 2 s + 8=s^(4) \left(A + C + D + E\right) + s^(3) \left(4 A + B + 3 C + 5 D\right) + s^(2) \left(- A + 4 B - 4 C + 4 D - E\right) + s \left(- 4 A - B\right) - 4 B`

Solve for each constant

`\begin{cases} A + C + D + E = 0\\4 A + B + 3 C + 5 D = 0\\- A + 4 B - 4 C + 4 D - E = 5\\- 4 A - B = 2\\- 4 B = 8 \end{cases}`

`-4B=8\\B=-2`

`-4A-B=2\\-4A-(-2)=2\\-4A+2=2\\-4A=0\\A=0`

`A+C+D+E=0\\C+D+E=0\\E=-C-D`

`-A+4B-4C+4D-E=5\\4(-2)-4C+4D-(-C-D)=5\\-8-4C+4D+C+D=5\\-3C+5D=13\\5D=13+3C`

`4A+B+3C+5D=0\\4(0)+(-2)+3C+13+3C=0\\-2+6C+13=0\\11+6C=0\\6C=-11\\C=-(11)/(6)`

`5D=13+3C\\5D=13+3(-(11)/(6))\\5D=13-(33)/(6)\\5D=(15)/(2)\\D=(15)/(10)\\D=(3)/(2)`

`E=-C-D\\E=-(-(11)/(6))-((3)/(2))\\E=(11)/(6)-(3)/(2)\\E=(2)/(6)\\E=(1)/(3)`

Take the inverse transform and solve for the IVP

`Y(s)=(0)/(s)+(-2)/(s^2)+(-(11)/(6))/(s+1)+((3)/(2))/(s-1)+((1)/(3))/(s+4)\\ \\y(x)=-2x-(11)/(6)e^(-x)+(3)/(2)e^x+(1)/(3)e^(-4x)`