# Help with second order initial value problem

• Mar 8th 2010, 05:32 PM
steph3824
Help with second order initial value problem
$\displaystyle y''+8y'-9y=0, y(1)=1, y'(1)=0$

I have found that the two roots for this are $\displaystyle r_1=-9, r_2=1$

So, $\displaystyle y(t)=c_1e^{-9t}+c_2e^t$
$\displaystyle y'(t)=-9c_1e^{-9t}+c_2e^t$

After this, I get pretty lost trying to get the correct values for c1 and c2. Can someone help out with this part please? I know that the ans. to this problem is $\displaystyle y=1/10e^{9-9t}+9/10e^{-1+t}$
• Mar 8th 2010, 05:41 PM
pickslides
Quote:

Originally Posted by steph3824
$\displaystyle y''+8y'-9y=0, y(1)=1, y'(1)=0$

So, $\displaystyle y(t)=c_1e^{-9t}+c_2e^t$
$\displaystyle y'(t)=-9c_1e^{-9t}+c_2e^t$

Use $\displaystyle y(1)=1, y'(1)=0$

So,

$\displaystyle y(1)=c_1e^{-9\times 1}+c_2e^1 = 1$

and

$\displaystyle y'(1)=-9c_1e^{-9\times 1}+c_2e^1= 0$

Now simplify and solve.
• Mar 9th 2010, 05:57 PM
steph3824
Well that's what I'm needing help with is the simplifying. I think that $\displaystyle c_2=1-c_1e^{-9}/e$ but I'm not even positive this is correct, does it become $\displaystyle c_2=1-c_1e^{-10}$?

So if the first thing is correct you get $\displaystyle -9c_1e^{-9}+(1-c_1e^{-9}/e)e=0$ and then $\displaystyle -9c_1e^{-9}+1-c_1e^{-9}=0$

I don't really know if that is correct and if it is how do you get c_1? I've forgotten the exponent rules and what not
• Mar 9th 2010, 06:02 PM
pickslides
I get $\displaystyle c_2 = \frac{-9c_1}{e^{10}}$
• Mar 9th 2010, 06:03 PM
steph3824
Nevermind, after working with the numbers a bit I have figured it out :)