# 2nd order ODE with a catch

• Jun 15th 2011, 03:11 PM
Lancet
2nd order ODE with a catch
I have an odd ODE question that I could use a hand with...

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$\displaystyle y'' - y' - 2y = 0$

$\displaystyle y(0) = a$

$\displaystyle y'(0) = 2$

Find "a" so that the solution approaches 0 as t approaches infinity.

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So here's what I've done:

$\displaystyle r^2 - r - 2 = 0$

$\displaystyle r = -1, 2$

$\displaystyle y = C_{1}e^{-t} + C_{2}e^{2t}$

$\displaystyle a = C_{1} + C_{2}$

$\displaystyle C_{2} = a - C_{1}$

$\displaystyle y' = -C_{1}e^{-t} + 2C_{2}e^{2t}$

$\displaystyle 2 = -C_{1} + 2C_{2}$

$\displaystyle C_{1} = 2C_{2} - 2$

$\displaystyle C_{1} = 2a - 2C_{1} - 2$

$\displaystyle C_{1} = \frac{2a - 2}{3}$

$\displaystyle y = \frac{C_{1}}{e^t} + C_{2}e^{2t}$

So, unless I've made a mistake somewhere, I've done most of the work. But I don't understand how I can get an "a" that will cause the solution to approach 0 as t approaches infinity. Since the second term is not part of a fraction, I'm not sure how this can be done.

Can someone help me to understand what I'm missing?
• Jun 15th 2011, 03:33 PM
mr fantastic
Re: 2nd order ODE with a catch
Quote:

Originally Posted by Lancet
I have an odd ODE question that I could use a hand with...

----------------

$\displaystyle y'' - y' - 2y = 0$

$\displaystyle y(0) = a$

$\displaystyle y'(0) = 2$

Find "a" so that the solution approaches 0 as t approaches infinity.

----------------

So here's what I've done:

$\displaystyle r^2 - r - 2 = 0$

$\displaystyle r = -1, 2$

$\displaystyle y = C_{1}e^{-t} + C_{2}e^{2t}$

$\displaystyle a = C_{1} + C_{2}$

$\displaystyle C_{2} = a - C_{1}$

$\displaystyle y' = -C_{1}e^{-t} + 2C_{2}e^{2t}$

$\displaystyle 2 = -C_{1} + 2C_{2}$

$\displaystyle C_{1} = 2C_{2} - 2$

$\displaystyle C_{1} = 2a - 2C_{1} - 2$

$\displaystyle C_{1} = \frac{2a - 2}{3}$

$\displaystyle y = \frac{C_{1}}{e^t} + C_{2}e^{2t}$

So, unless I've made a mistake somewhere, I've done most of the work. But I don't understand how I can get an "a" that will cause the solution to approach 0 as t approaches infinity. Since the second term is not part of a fraction, I'm not sure how this can be done.

Can someone help me to understand what I'm missing?

If y --> 0 as t --> +oo then C2 has to equal zero.

So you have y = C1 e^(-t) and since y(0) = a, C1 = a.

So y = a e^(-t).

Now use y'(0) = 2 to get a.
• Jun 15th 2011, 04:03 PM
Lancet
Re: 2nd order ODE with a catch
Quote:

Originally Posted by mr fantastic
If y --> 0 as t --> +oo then C2 has to equal zero.

So you have y = C1 e^(-t) and since y(0) = a, C1 = a.

So y = a e^(-t).

Now use y'(0) = 2 to get a.

Thank you so much for that - I just couldn't see what I was missing. In retrospect, it seems obvious, and worth kicking myself over. :)

Thanks again!
• Jun 16th 2011, 03:40 AM
HallsofIvy
Re: 2nd order ODE with a catch
No, no, don't kick yourself. Let us do it!