- #1

- 2

- 0

## Homework Statement

Find all solutions to [dx/dt; dy/dt] = [1, 2; 0, 1]*[x; y]

## Homework Equations

the eigenvalue characteristic equation: det(A-λ*I)=0

## The Attempt at a Solution

This results in real, repeated eigen values: λ1,2 = 1

for λ1 = 1, (1-1)k1 + 2k2 = 0

choose k1 = 1 then k2 = 0 gives us the eigenvector K1 = [1, 0]

this gives part of the general solution X1 = [1, 0]*e^t

to get a second solution we create a new vector P to avoid duplication

with K1 = [1,0] and P = [p1; p2] we solve (A-λ2*I)*P=K

(1-1)*p1 + 2*p2 = 1

choose p1 = 0 then p2 = 1/2

this gives us the solution X2 = [1;0]*t*e^t + [0;1/2]*e^t

putting it together X = X1 + X2

[x; y] = c1*[1, 0]*e^t + c2*([1;0]*t*e^t + [0;1/2]*e^t)

or

x = c1*e^t + c2*t*e^t

y = (1/2)*c2*e^t

Have I found all the solutions to this system? I think this is the general solution, is this all they are asking for?

If we multiply an eigenvalue by some multiple to get a new eigenvalue, is it really possible to find all the solutions? It seems like there would be infinitely many..