Simple homogenous 2nd order Diff EQ, but i'm stuck on the process

[mr_coffee]

Hello everyone, we just started 2nd order differentials, and i was loooking at his example and it made senes but now I'm doing the homework and I'm stuck.

Here is the problem:
Find y as a function of t if
y'' - 3y' = 0,
y(0) = 9, y(1) = 7 .
y(t) =?

Well there is my work
http://img135.imageshack.us/img135/495/lastscan6vi.jpg

 

[benorin]

you have it right, just need to solve for c1 and c2 from this system (same as you have):

c_1+c_2=9
c_1+c_2e^3=7

 

[benorin]

Use matrix methods: the system is

\left(\begin{array}{cc}1&1\\1&e^3\end{array}\right) \left(\begin{array}{cc}c_1\\c_2\end{array}\right) =\left(\begin{array}{cc}9\\7\end{array}\right)

 

[benorin]

multipling on the left by the inverse of the coefficient matrix gives

\left(\begin{array}{cc}c_1\\c_2\end{array}\right) = \frac{1}{(1)(e^3)-(1)(1)} \left(\begin{array}{cc}e^3&-1\\-1&1\end{array}\right) \left(\begin{array}{cc}9\\7\end{array}\right) = \frac{1}{e^3-1} \left(\begin{array}{cc}9e^3-7\\-9+7\end{array}\right) = \left(\begin{array}{cc}\frac{9e^3-7}{e^3-1}\\-\frac{2}{e^3-1}\end{array}\right)

so c_1=\frac{9e^3-7}{e^3-1}\mbox{ and }c_2=-\frac{2}{e^3-1}