# find the general solution

• Apr 7th 2009, 02:19 PM
mandy123
find the general solution
Find the general solution using the method of elimination
x'=x+2y+z
y'=6x-y
z'=-x-2y-z

I used a matrix and found
D^3 + D^2 - 12D=0
and then I think
x(t)= A + Be^(-4t) +Ce^(3t)

But then I don't know where to go, or how to solve it,

Thanks
• Apr 8th 2009, 03:27 PM
Jester
Quote:

Originally Posted by mandy123
Find the general solution using the method of elimination
x'=x+2y+z
y'=6x-y
z'=-x-2y-z

I used a matrix and found
D^3 + D^2 - 12D=0
and then I think
x(t)= A + Be^(-4t) +Ce^(3t)

But then I don't know where to go, or how to solve it,

Let's do it a little different. First solve the second equation for $\displaystyle x$ so $\displaystyle x = \frac{y'+y}{6}\;\;(1)$. Substitute this into the first equation. This gives
$\displaystyle \frac{y''+y'}{6} = \frac{y'+y}{6} + 2y + z$. Solving for $\displaystyle z$ gives $\displaystyle z = \frac{y'' - 13y}{6}\;\;(2)$. Now substitute this into the first equation giving (after simplying gives
$\displaystyle y''' + y'' - 12y' = 0$ (same equation as for x)
The solution as you said is $\displaystyle y = c_1 + c_2 e^{-4t} + c_3e^{3t}$ (I've chosen different constants not to be confusing with your x(t) solution). Now substitute this into (1) and (2) above to get $\displaystyle x\; \text{and}\; z$.