- #1
nut.the.dutch
- 7
- 0
Greetings all/any:
Take the Laplace transform of the following initial value problem and solve for Y(s) = L{y(t)}
y'' - 3y' -28y = 1 if 0<t<1
0 if 1<t
initial conditions: y'(0) = y(0) = 0
a. Y(s) = ?
b. now find the inverse transform to find y(t). Use step(t-c) for uc(t)
*Note:
1/(s(s-7)(s+4)) = (-1/28)/s + (1/44)/(s+4) + (1/77)/(s-7)
not entirely sure. I know random equations for the left hand side of the shifted equation, but I'm lost as to how to even start applying them. Nevertheless:
L{f(t-a) u(t-a)} = F(s)e-as
L{g(t) - g(t-a)f(t-a)}
...etc? again, I have all of these equations, but I don't know how to use them so even if I found out some way to just plug numbers in, I still wouldn't understand what the concept behind this is.
I can solve the left hand side:
y'' - 3y' -28y = f(t); y(0)=y'(0)=0
-3L{y'(t)} = sL{y(t)} - y(0) = -3(sY(s) - 0) = -3sY(s)
L{y''(t)} = s2L{y(t)} -sy(0) -s'(0) = s2Y(s) - 0s - 0 = s2Y(s)
= [s2Y(s)] -3[sY(s)] - 28[Y(s)] = F(s)
= Y(s) [s2 -3s -28]= F(s)
...and then I'd have to find the laplase tranform of the right hand side as well, and divide through to isolate and solve for Y(s)
That said. I have no idea where to start on the right hand side. I've looked at the book, (Edwards & Penny 4th ed.), looked at notes, tried finding stuff online - basically what I get is a lot of examples of solutions, but I don't understand what they're doing to find their right hand side equations...
Thank you.
Homework Statement
Take the Laplace transform of the following initial value problem and solve for Y(s) = L{y(t)}
y'' - 3y' -28y = 1 if 0<t<1
0 if 1<t
initial conditions: y'(0) = y(0) = 0
a. Y(s) = ?
b. now find the inverse transform to find y(t). Use step(t-c) for uc(t)
*Note:
1/(s(s-7)(s+4)) = (-1/28)/s + (1/44)/(s+4) + (1/77)/(s-7)
Homework Equations
not entirely sure. I know random equations for the left hand side of the shifted equation, but I'm lost as to how to even start applying them. Nevertheless:
L{f(t-a) u(t-a)} = F(s)e-as
L{g(t) - g(t-a)f(t-a)}
...etc? again, I have all of these equations, but I don't know how to use them so even if I found out some way to just plug numbers in, I still wouldn't understand what the concept behind this is.
The Attempt at a Solution
I can solve the left hand side:
y'' - 3y' -28y = f(t); y(0)=y'(0)=0
-3L{y'(t)} = sL{y(t)} - y(0) = -3(sY(s) - 0) = -3sY(s)
L{y''(t)} = s2L{y(t)} -sy(0) -s'(0) = s2Y(s) - 0s - 0 = s2Y(s)
= [s2Y(s)] -3[sY(s)] - 28[Y(s)] = F(s)
= Y(s) [s2 -3s -28]= F(s)
...and then I'd have to find the laplase tranform of the right hand side as well, and divide through to isolate and solve for Y(s)
That said. I have no idea where to start on the right hand side. I've looked at the book, (Edwards & Penny 4th ed.), looked at notes, tried finding stuff online - basically what I get is a lot of examples of solutions, but I don't understand what they're doing to find their right hand side equations...
Thank you.