- #1
dmoney123
- 32
- 1
Homework Statement
Consider the initial value problem for the system of first-order differential equations
y_1' = -2y_2+1, y_1(0)=2
y_2' = -8y_1+2, y_2(0)=-1
If the matrix
[ 0 -2
-8 0 ]
has eigenvalues and eigenvectors L_1= -4 V_1= [ 1
2 ]
L_2=4 V_2= [ 1
-2]
then its solution will be:
Homework Equations
The Attempt at a Solution
[/B]
e^(-4t) +e^ (4t) from eigenvalues
multiply by respective eigenvectors and set to initial conditions gives 2 sets of equations and two unknown coefficients
2=c_1*e^(-4t)+c_2*e^(4t)
-1=c_1*2e^(-4t)+c_2*-2e^(4t)
c_1=3/4
c_2=5/4
I am very confident with these values being right for the coefficients, I know need to know how to use these to form a general solution.
I plug these values back into get
y_1=3/4e^(-4t)+5/4e^(4t)
y_2=3/2e^(-4t)-5/2e^(4t)
Then solution given is
y_1(t)=5/4e^(4t)+1/2e^(-4t)+1/4,
y_2(t)=-5/2e^(4t)+e^(-4t)+1/2
any help is appreciated! thanks!