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Laplace transforms, Inverse Laplace transforms, & Diff Eqns

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Please walk me through these problems step by step.

1. Compute the Laplace transform of e^(-10t) = u(t).

2. Compute the inverse Laplace transform of:

X(s) = (3s^2+2s+1) / (s^3+5s^2+8s+4)

3. Use Laplace transforms to compute the solution to the differential equation given in the attachment.

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https://brainmass.com/engineering/electrical-engineering/laplace-transforms-inverse-laplace-transforms-diff-eqns-367572

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See Also This Related BrainMass Solution

Differential Equation and Matrix Determinant

1.The ODE X"+ kX = 0 has different types solution depending on sign of k.
We consider the three possible cases separately.
k=0: X"=0 so that X(x)=Ao+Box, X'(0)=0 implies Bo=0 so that X(x)=Ao. Finally, X(1)=0 implies Ao=0 and there is only trivial solution X=0.

The matrix has determinant e^mu + e^-mu= 2cosh.mu, which is never zero and it follows that pair of equations has unique solution Ahat=Bhat=0 and only trivial solution X=0.
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