1. ## Systems of ODEs

Using the Laplace transform, solve the initial value problem:
y1’ = -6y1 + 4 y2 (1)
y2’ = -4y1 + 4 y2 (2)
y1(0) = -2
y2(0) = -7

Solving:
(1) becomes: sY1 –y1(0) = -6Y1 + 4Y2 => sY1 +2 = -6Y1 + 4Y2 (3)
(2) becomes: sY2 – y2(0) = -4Y1 + 4Y2 => sY2 + 7 = -4Y1 + 4Y2 (4)

Two equations with 2 unknowns, so starting with (4), we get:
Y1 = [7-(4-s)Y2]/-4

Plugging into (3), we get:
Y2 = [7s+50]/[(s+4)(s-2)]

At this point, I can use partial fractions to determine A/(s+4) and B/(s-2), but I can’t help but think I messed up the math somewhere as I can’t match the stated answer of:
y1(t) = 2e**-4t – 4e**2t
y2(t) = e**-4t – 8e**2t
The (s+4) and (s-2) terms of my answer makes me think I’m tantalizingly close, but any help on where I messed up will be much appreciated.

2. Originally Posted by dsprice
Using the Laplace transform, solve the initial value problem:
y1’ = -6y1 + 4 y2 (1)
y2’ = -4y1 + 4 y2 (2)
y1(0) = -2
y2(0) = -7

Solving:
(1) becomes: sY1 –y1(0) = -6Y1 + 4Y2 => sY1 +2 = -6Y1 + 4Y2 (3)
(2) becomes: sY2 – y2(0) = -4Y1 + 4Y2 => sY2 + 7 = -4Y1 + 4Y2 (4)

Two equations with 2 unknowns, so starting with (4), we get:
Y1 = [7-(4-s)Y2]/-4

Plugging into (3), we get:
Y2 = [7s+50]/[(s+4)(s-2)]

At this point, I can use partial fractions to determine A/(s+4) and B/(s-2), but I can’t help but think I messed up the math somewhere as I can’t match the stated answer of:
y1(t) = 2e**-4t – 4e**2t
y2(t) = e**-4t – 8e**2t
The (s+4) and (s-2) terms of my answer makes me think I’m tantalizingly close, but any help on where I messed up will be much appreciated.
The error is in:

Plugging into (3), we get:
Y2 = [7s+50]/[(s+4)(s-2)]

this is wrong. It should be:

$Y_2(s) = -\;\frac{7s+34}{(s+4)(s-2)}$

Which I believe gives:

$y_2(t)=e^{-4t}-8e^{2t}$

CB

3. ## Systems of ODEs

Drats! And I even checked my math at least 10 times! Thanks. But now when I try to solve for Y1, I get:

Y1 = [-2s – 18]/[(s+4)(s-2)]

This doesn’t seem quite right. And I’ve once again checked my math at least 10 times and am running out of paper. J

4. Originally Posted by dsprice
Drats! And I even checked my math at least 10 times! Thanks. But now when I try to solve for Y1, I get:

Y1 = [-2s – 18]/[(s+4)(s-2)]

This doesn’t seem quite right. And I’ve once again checked my math at least 10 times and am running out of paper. J
it is:

$Y_1(s)=-\;\frac{2s+20}{(s+4)(s-2)}$

CB

5. Thanks again. It still took me several attempts to finally get the same result. I seem to be making math more difficult than it needs to be!