Laplace transformation to solve intial value problem

[Karmel]

Homework Statement

laplace transformation to solve intial value problem
y"-2y+y=-4e^1-t + 2t
y(1)=0, y'(1)=-3

Homework Equations

The Attempt at a Solution

K. I have one more on the review that I am unaware of where I need to start. So if anyone can point me in the right direction please feel free. My graduation depends on the final and I am so lost on how to start amny of these problems... Thanks in advance

 

[rock.freak667]

Take the Laplace transformation on both sides noting that L{y(t)}=Y(s), then just make Y(s) the subject, then find the inverse laplace transform of Y(s) and you will the solution.

 

[Karmel]

okay so I found the laplace of both sides and get

2/s^3 + 2/s^2 + 1/s = (-4e(-t+1)/s) + 2t/s

I'm not really sure where to go form here. I think that I want to set
Y=L{y(t)}(s)
L{y'}=sY
L{y"}=s^2Y

but I am not so sure that is the direction I need to head and I really am not sure were all this fits into the problem. Where do I head from here?

 

[rock.freak667]

L{y(t)}=sY(s)-Y'(0)

L{y''(t)}=s²Y(s)-sY(0)-Y'(0)

L{e^kt}= 1/(s-k)

L{tⁿ}=n!/sⁿ⁺¹

 

[Karmel]

ok I get the first two but why the second two in the above reply. I guess I don't know what I need to do with them...

 

[rock.freak667]

On looking back to the question, you need to change the variable such that the conditions occur when t=0. To do this we let t=w+1

so now we have

y''(w-1)-2y'(w-1)+y(w+1)=4e^-1-w-1+2(w+1)

Now define u(w)=y(w+1)
so now u'(w)=y'(w+1) and u''(w)=y''(w+1)

Now when w=0 => u(0)=y(0+1)=y(1)=0 AND u'(w)=y'(1)=-3

so now you need so solve the new equation

u''-2u'+u=4e^-w+2w+2


Now take the Laplace transform on both sides of the equation.

to elaborate on the second two...


if you need to get L{e^5t} it is simplt 1/(s-5)

and L{t²}=2!/s³

 

[Karmel]

Okay so now I get
L{u"}(s) = s2U(s) + 3
L{u'}(s) = sU(s)


u''-2u'+u=4e-w+2w+2

taking the LT of both sides I get
s2U(s)+3-2sU(s)+U(s)=(4/s+1)+(2/s2)+(2/s)
(s2-2s+1)U(s)=(4/s+1)+(2/s2)+(2/s)-3

I feel like I have messed up somewhere cause if I solve for U(s) then it gets really ugly...