SOLUTION: Hello, looking for a bit of guidance with Laplace transforms. Could I please be shown how to: 1. Use the Laplace transform to solve for X: 8 dx/dt + 4 given x=0 @ t=0 2. U

Question 970844: Hello,
looking for a bit of guidance with Laplace transforms. Could I please be shown how to:
1. Use the Laplace transform to solve for X: 8 dx/dt + 4 given x=0 @ t=0
2. Use the Laplace Integral: L[f(t)]=F(s)=∫_0^∞ e^(-st) f(t)dt to find the transform of the voltage shown in an exponential decay for t≥0.
Thanks a lot in advance.
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
1) solve for X: 8 dx/dt + 4 given x=0 @ t=0
I assume you mean 8 * (dx/dt) + 4 = 0
*******************************************
The general rule of Laplace transforms is
L[x'] = sL[x(t)] - x(0)
So taking the laplace transform of both sides
**********************************************
L[8 * x' + 4] = L[0]
L[8x'] + L[4] = L[0]
note that L[c] = c/s and L[0] = 0
8L[x'] + 4/s = 0
8*(sL[x(t)] - x(0)) + 4/s = 0
since we are given x(0) = 0
8sL[x(t)] + 4/s = 0
divide both sides of = by 4
2sL[x(t)] + 1/s = 0
2sL[x(t)] = -1/s
divide both sides of = by s
2L[x(t)] = -1/(s^2)
divide both sides of = by 2
L[x(t)] = -1/(2*s^2)
************************************************
2) L[f(t)]=F(s)=∫_0^∞ e^(-st) f(t)dt to find the transform of the voltage shown in an exponential decay for t≥0.
voltage shown?

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