Can anyone answer this problems using Laplace Transform and Partial Fraction Expansion?

1.) d^2x/dt^2 + 6dx/dt + 8x = 5sin3t
2.) d^2x/dt^2 + 8dx/dt + 25x = 10 u(t)
3.) d^2x/dt^2 + 2dx/dt + x = 5e^-2t

Thanks.

Well, have you made an attempt at the answer? We can't just do your homework for you.

Well, have you made an attempt at the answer? We can't just do your homework for you.

Yes of course, I have used 3 papers trying to answer them.

Here are my answers:

1.) my answer is probably wrong

2.) -10/25 e^-4t cos3t -4/15 e^-4t sin3t - 10/25

3.) 5 e^-2t + 5te^-t - 5e^-t (Probably Correct)

Do you have initial conditions you forgot to post? LaPlace usually comes with initial conditions. Just checking. y(0)=? y'(0)=? for example.

Oh yes, I'm sorry, I forgot, the initial conditions are y(0)=0, y'(0)=0

But I think that the initial conditions is used on the "Direct Method", while Laplace Transform don't need initial condition.. I'm not sure actually, any opinion would be greatly appreciated.

I will help with the first one. You see if you can finish, OK?

x''+6x'+8x=5sin(3t), \;\ y(0)=0, \;\ y'(0)=0

Use Laplace subs:

p^{2}Y-py(0)-y'(0)+6(pY-y(0))+8Y=\frac{15}{p^{2}+9}

Factor out and solve for Y:

Y(p^{2}+6p+8)=\frac{15}{p^{2}+9}

Y=\frac{15}{(p^{2}+9)(p^{2}+6p+8)}

Partial fractions:

Y=\frac{-18p}{65(p^{2}+9)}-\frac{3}{65(p^{2}+9)}-\frac{3}{10(P+4)}+\frac{15}{26(p+2)}

Now, use the inverse Laplace that correspond with the respective elements of the partial fraction decomposition. . I assume you have the table. That's what I use.

Write back if you're still stuck.