Laplace Transform Problem - Peacewise Functions

[bathtub2007]

Okay so this problem has been giving me nightmares ever since I laid eyes on it.


The problem: y'' - y = f(t), y(0) = 1, y'(0) = 0

f(t) = 1, 0 (less than equal to) t (less than) pi/2
f(t) = sin(t), t (greater than equal to) pi/2



Any help would be appreciated putting me in the right direction.

Thank you in advance.


--Julie

 

[xfunctionx]

I think the best way forward would be:

 

[matticus]

except f(t) isn't 1, read the post again. for piecewise functions use the unit step (heaviside) function.

 

[djeitnstine]

You can rewrite f(t) in terms of the heaviside function as matticus said, then perform the transform.

$$f(t)=1+H(t-\frac{\pi}{2})+\left(sin(t)-1\right)H(t-\frac{\pi}{2})$$

Edit since latex isn't working...

f(t)=1+H(t-[pi/2])+[sin(t)-1]*H(t-[pi/2])

where H(t-[pi/2]) denotes the unit step function.

 

[Maths.tuition]

Hi I was thinking about the same approach as xfunctionx.

Please advise to see whether his approach is correct or not.

Thanks

 

[djeitnstine]

maths.tuition welcome to PF.

If you are speaking of doing the same thing that xfunctionx did for bathtub's differential equation that would be wrong

f(t) = 1, 0 (less than equal to) t (less than) pi/2
f(t) = sin(t), t (greater than equal to) pi/2

does not simply equal 1 for all t, please note I already stated the proper form of the function(in terms of the unit step function) to get the the laplace transform for f(t)

"f(t)=1+H(t-[pi/2])+[sin(t)-1]*H(t-[pi/2])"

 

[coomast]

maths.tuition welcome to PF.

If you are speaking of doing the same thing that xfunctionx did for bathtub's differential equation that would be wrong

f(t) = 1, 0 (less than equal to) t (less than) pi/2
f(t) = sin(t), t (greater than equal to) pi/2

does not simply equal 1 for all t, please note I already stated the proper form of the function(in terms of the unit step function) to get the the laplace transform for f(t)

"f(t)=1+H(t-[pi/2])+[sin(t)-1]*H(t-[pi/2])"

I think this last expression is wrong. Musn't this be:

$$L\left{H(t)-H\left(t-\frac{\pi}{2}\right)+sin(t)H\left(t-\frac{\pi}{2}\right)\right}= L\left{H(t)-H\left(t-\frac{\pi}{2}\right)+cos\left(t-\frac{\pi}{2}\right)H\left(t-\frac{\pi}{2}\right)\right}$$

There seems to be something definitely wrong with the latex generation, it keeps posting things from older replies of other posts of me :grumpy:

coomast

 

[djeitnstine]

You are right the correct function is

f(t)=1+H(t-[pi/2])+[sin(t)-2]*H(t-[pi/2])

one had to subtract 2 from sin and not 1 or else you would be left with 2-1+sin{t} = 1+sin(t)

thanks for catching that mistake

 

[coomast]

You are right the correct function is

f(t)=1+H(t-[pi/2])+[sin(t)-2]*H(t-[pi/2])

one had to subtract 2 from sin and not 1 or else you would be left with 2-1+sin{t} = 1+sin(t)

thanks for catching that mistake

OK, next attempt for a good latex formula. I think it is still not OK, I got:

$$L\left[H(t)-H\left(t-\frac{\pi}{2}\right)+sin(t)H\left(t-\frac{\pi}{2}\right)\right]= L\left[H(t)-H\left(t-\frac{\pi}{2}\right)+cos\left(t-\frac{\pi}{2}\right)H\left(t-\frac{\pi}{2}\right)\right]$$

coomast