Laplace Transform : sin^2(wt) question
Hello! I am working on my Laplace Transform homework and I am trying to work out the steps on how my professor got this answer but I am not understanding it. The question is:
Q: What is the inverse Laplace transform of {32 / (s(s[SUP]2[/SUP] + 64))} ?
A: sin[SUP]2[/SUP](4t)
The Laplace Look Up Table formula is :
I really appreciate it if someone can help me understand how the answer is sin[SUP]2[/SUP](4t)
Q: What is the inverse Laplace transform of {32 / (s(s[SUP]2[/SUP] + 64))} ?
A: sin[SUP]2[/SUP](4t)
The Laplace Look Up Table formula is :
Signal Transform sin[SUP]2[/SUP](w[SUB]0[/SUB]t) 2w[SUB]0[/SUB][SUP]2[/SUP] / (s(s[SUP]2[/SUP] + 4w[SUB]0[/SUB][SUP]2[/SUP]))When I am breaking down this function and I am using the transform equation I can only break down the 64 (since 8[SUP]2[/SUP] = 64) but that 4 right next to it and that 32 is throwing me for a loop.
I really appreciate it if someone can help me understand how the answer is sin[SUP]2[/SUP](4t)
Replies

Sagar07Didn't get your question exactly.......

Brad Grillakis
My question is, from this function 32/(s(s[SUP]2[/SUP]+64)), how is the answer sin[SUP]2[/SUP](4t)? (I have been trying to work out how my professor got this answer but I can not seem to figure this out. I tried doing partial fractions, but u cant because of the s[SUP]2[/SUP].)Sagar07Didn't get your question exactly.......
The 2 formula I gave is from the Laplace Transform Table. 
ritesh05Here as we can see that in the denominator there is 's' term so we can directly use the division by s formula that is
so after using this formula you will get .5(cos8t1)=.5(12sin^(4t)1)=sin^2(4t)...............{cos2t=12sin^2(t)} 
rashidsohaili dont understand that
Here as we can see that in the denominator there is 's' term so we can directly use the division by s formula that is 
simplycoder
This can be solved by partial fractions. All you need is proper substitution.Brad GrillakisMy question is, from this function 32/(s(s[SUP]2[/SUP]+64)), how is the answer sin[SUP]2[/SUP](4t)? (I have been trying to work out how my professor got this answer but I can not seem to figure this out. I tried doing partial fractions, but u cant because of the s[SUP]2[/SUP].)
The 2 formula I gave is from the Laplace Transform Table.
Substitute it as (A/s)+(Bs+C)/(s^2+64).
A=0.5,B=0.5,C=0.
so you get 0.5/s 0.5/(s^2+64)
Laplace inverse of the first term is 1, and that of second term is cos(8t).
So you get 0.5*(1cos8t).By trignometry, cos8t=12sin^2 (4t).
So you get 0.5*(1(12sin^2(4t))).
=0.5*2sin^2(4t).
=sin^2(4t).// 
silverscorpionWrite the fraction 32/(s(s2+64) as follows..
32/(s(s2+64) = (2*4[sup]2[/sup])/(s*(s[sup]2[/sup] + 4*4[sup]2[/sup]))
So, you get w[sub]0[/sub] to be 4. Hence, the inverse Laplace transform is, from the formula you have given, sin[sup]2[/sup] 4t
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