Hi,
I have a couple of more that I did. Could someone please check and see if these are right.
1. FIND THE LAPLACE Transform of the unknown solution function for the following initial value problem:
y" + 4y' - 5y = te^(t), y(0) = 1, y'(0)=0
(Do Not actually find the function, only its transform. Then, without carrying out the steps, inidicate briefly how you would proceed to find the solution function.)
s^2L = s + 4sL - 4 - 5L = Lte^t = 1/(s-1)^2
(s^2 + 4s - 5)l = 1/(s-1)^2 + s +4 = (s^3+ 2s^2-7s+4)/(s-1)^2
simplify
= (s-1)^2(s+4)/(s-1)^3(s+5) = s=4/(s-1)(s+5)
at this point I would do partial fractions.
2.Find Laplace Inverse
L^-1 { (s+10)/(s^2+8s+20)}
s+10)/(s^2+8s+20) = (s+10)/(s^2+8s+16+4) = (s+10)/((s+4)^2+4)
(s+10)/((s+4)^2+4) = (s-(-4))+ 3*2)/(s-(-4))^2+2^2) = (s-(-4)/((s-(-4))^2+2^2) + 3*2/(s-(-4)+2^2)
The Inverse Laplace Transform of that is
e^-4t cos 2t + 3 e^-4t sin 2t.
3. Find Laplance Transform
Find L (f(t)} where f(t) = e^-t (0 <= t < 5)
-1 (t >= 5)
For t in [0,5) the solution would be Int((e^{-t})(e^{-st},0,infinity,dt)=Int(e^{-t(s+1)},0,infinity,dt}=-e^{-t(s+1)}/(s+1)|0,infinity) = 1/(s+1).
For t in [5,infinity) Int(-1*e^{-st},0,infinity,dt} =(1/s)e^{-st}|0,infinity)=-1/s.
In piecewise notation you write L(f(t))={1/s, 0 <= t < 5 and -1/s, t >= 5}
4. Find Laplace Inverse of
L^-1 { (3s-4)/s(s-4) }
so
3s-4)/s(s-4) = A/s + B/(s-4)
s*(3s-4)/(s(s-4)) = A + B*s/(s-4)
Taking limit ( s to 0)
-4/-4 = A
A =1
And
(s-4)*(3s-4)/(s(s-4)) = A*(s-4)/s + B
Taking limit ( s to 4)
8/4 = B = 2
so
(3s-4)/s(s-4) = 1/s + 2/(s-4)
Thank you