Question 997172: Hi
Really stuck on a problem here. If somone could help me out would be greatly appreciated
Find the solution of the following system of differential equations:
x'1 = x1 + x2
x'2 = x1 + x2
With the initial values
x1(0) = 1, x2(0) = 2
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! The first step is to convert this system to matrix form
| x'1 | = | x1 + x2 |
| x'2 | = | x1 + x2 |
note the right side of the = can be written as matrix multiplication
| x'1 | = | 1 1 | | x1 |
| x'2 | = | 1 1 | | x2 |
now we an define
x- =
| x1 |
| x2 |
x'- =
| x'1 |
| x'2 |
there we can write
x'- =
| 1 1 | x-
| 1 1 |
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now we calculate the eigenvalues, let r1 and r2 represent eigenvalues
for any 2 by 2 matrix
| a b |
| c d |
its characteristic function is
r^2 - (a+d)r + (ad-bc) = 0
for our problem we get
r^2 - 2r + 0 = 0, then
r1 = 0 or r2 = 2 these are our eigenvalues
*********************************************************************
let k1 and k2 be the associated eigenvectors
for r = 0, the matrix (A - rI) =
| 1 1 |
| 1 1 | and
k1 = (1 -1)
for r = 2, the matrix (A - rI) =
| -1 1 |
| 1 -1 | and
k2 = (1 1)
note we get the eigenvectors by picking row one and putting a - sign in front of the first element of the first row and then transposing the elements in the first row
*********************************************************************
the general solution has the following equation for x
x = (C1 * k1 * e^(r1*t)) + (C2 *k2 * e^(r2*t))
for our problem we have
x = ( C1 * (1 -1) * 1 ) + ( C2 * (1 1) * e^(2t) )
note the initial condition is x(0) = ( 1 2 )
x(0) = ( C1 * (1 -1) ) + ( C2 * (1 1) ) = (1 2)
that is
C1 + C2 = 1
-C1 + C2 = 2
then C1 = -0.5, C2 = 1.5
**********************************************************************
therefore our solution is
x = ( -0.5 * (1 -1)) + ( 1.5 * (1 1)* e^(2t) )
x = ( ( -0.5 0.5 ) ) + ( (1.5e^2t 1.5e^2t) )
x = ( -0.5+1.5e^2t 0.5+1.5e^2t )
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