Solving a Laplace Transform Problem: Where Am I Going Wrong?

[metdave]

I'm out of college and am brushing up on Laplace Transforms. I have a problem I've solved, but I believe the solution I got is wrong and can't find my error.

The problem is 2x''-x'=t*sin(t) x(0)=5,x'(0)=3

My solution...

Take the Laplace Transform

2(s^2x-5s-3)-(sx-5)=2s/(s^2+1)^2

Rearranging, I get
x(2s^2-s)-10s-1=2s/(s^2+1)^2

Solve for x
x=(10s+1)/(2s^2-s)+2/((2s-1)(s^2+1)^2

Then, doing a PFD on the first term, I get -1/s+8/(2s-1)

Doing an inverse Laplace Transform, I get x(t)=-1+8e^(t/2)+Integral((sin(y)-ycos(y)(e^(1/2)((t-y))dy,0,y)

I used the convolution theorem on the second term on the RHS. That doesn't look right because the initial conditions aren't satisfied. Can anyone point me in the right direction?

Thanks!

 

[cryora]

To get inverse laplace of 1/(2s-1) I would rewrite as (1/2)/(s-1/2) which becomes (1/2)e^(1/2t). It appears you did not include the 1/2 factor for two of you terms.

 

[lurflurf]

Two things jump out
1)in 8/(2s-1) the 8 should be 332/25
2)The convolution should involve trigonometric functions not exponents
This rule is also useful here
$$\mathcal{L}^{-1} \{ \mathrm{F}(s) \} = t \, \mathcal{L}^{-1} \left\{ \int_s^\infty \! \mathrm{F}(u) \, \mathrm{d}u \right\}$$

 

[metdave]

Thanks for the help!

 

[blue_raver22]

As a scientist, it is important to carefully analyze and review your work to identify any potential errors. In this case, it seems that your initial Laplace Transform may have been incorrect. When solving for x, you should have used the initial conditions to solve for the constants, rather than rearranging the equation to isolate x. Additionally, it appears that your PFD may also be incorrect, which could explain why the initial conditions are not satisfied. I recommend double-checking your work and possibly seeking guidance from a professor or colleague to help identify any errors. It is important to remember that mistakes happen and seeking help is a valuable part of the learning process. Keep practicing and I am sure you will master Laplace Transforms in no time.