More Linear, Homogeneous Systems of Differential Equations

Once more we return to homogeneous linear systems of differential equations, to see how the solutions can be broken down and what we need to do. This time we will look at this example :

dx1/ dt = x1 + x2

dx2/dt = 4x1 + x2

  Note the first step, again, is to form a matrix from the coefficients, which we see are (1, 1) for the top, and (4, 1) for the bottom. Thus:   A =

(1 .....1)

(4......1)

Then, it must be true from the properties of determinants that:

(A - l) D =

[(1 - l).......1] [k1]

[4 ..... .(1 -l)] [k2]

Again, we allow l  to be subtracted from the first element in the upper left, and from the last element in the lower right). Cross-multiplying and using matrix properties we obtain the characteristic equation:

l² - 2l - 3 = 0



where l1 = 3 and l2 = -1 We need to first find a vector that solves the equation: (A - l1 I)D = 0
In the first instance, we substitute the first eigenvalue, l= 3, into the matrix, whence:

(A - 3I) D = 0 =

[-2.....1] [k1]

[4 ...-2] [k2]


Therefore:


-2k1 + k2 = 0, and  4k1 – 2k2 = 0,  So k1 = ½ and k2 = 1

Then our first eigenvector is: K1 =


[½]

[1]


Therefore, the first linearly independent solution for the system is:

X1 = K1 exp (l1 t) = K1 exp (3t) 

The second eigenvalue was l2 = -1 so we repeat the process again to obtain the equation to be solved:

(A - l2 I)D = (A - (-1)I) D = (A + I)D

Then, (A + I) D = 0 =


[2.....1] [k1]

[4 ….2] [k2]


Or: 2k1 + k2 = 0 and  4 k1 + 2k2 = 0

So that: k1 = 1, then k2 = - 2

The second eigenvector is then: K2 =


[1]

[-2]

 
So another linearly independent solution is:

X2 = K2 exp (-t)

Then add the two solutions, to obtain: 


X = X1 + X2 = K1 exp (3t) + K2 exp (-t)

With, of course, the 2 column vectors (as computed above) substituted in for K1, K2. This serves as a general approach for solving all such systems.
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Problems for the Math Maven:

1) Obtain the general solution(s) for the system:

dx/dt = 2x + y

 dy/dt = 2x + 3y

2) Obtain the general solution(s) for the system:

dx1/ dt = 3x1 - 4x2

dx2/dt = x1 - x2