Help required? Solve the Differential equation by the Laplace transform method"

Hello

I wonder if you can help? I believe I'm doing this correctly, but iv got caught up in the algebra. Ill list all the question and then as far as I have got.

Kind regards

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You are required to solve the following differential equation by the Laplace transform method:

d^2x/dt^2 + 5*dx/dt - 24x = 4t

with

x(0) = 2

x'(0)=9

(i)

Take the Laplace transform of the differential equation and solve the resulting equation for X, the Laplace transform of the solution

Give your answer as a function of only. Omit the "X ="

d^2x/dt^2 + 5*dx/dt - 24x = 4t

Taking the Laplace transform

L(d^2x/dt^2) + 5L(dx/dt) -24L (x) = 4L(t)

(s^2X - sx(0)-x'(0)) + 5( sX - x(0)) - 24X = 4/s^2

Sub in values given within questions for x(0) and x'(0)

(s^2X - 2s - 9) + (5sX - 10) - 24X = 4/s^2

Expanding brackets out and collecting like terms

X(s^2+5s-24) = 4/s^2 + 2s +19

Factorising LHS

X((s-3)(s+8)) = 4/s^2 + 2s +19

I can't see where to go next to, iv looked at brining the (s-3)(s+8) to the RHS but I get muddled with all the numbers and it doesn't seem to look correct.

Thanks again