DISTINCT EIGENVALUES -SYSTEM SOLVING:
Example
Solve the following IVP.
Solution:
Therefore, the first thing that we must to do that is, get the eigenvalues for the matrix.
= l^{2} -3l - 4
= (l+ 1) (l- 4) ⇒ l_{1} = -1, l_{2} = 4
Here let's get the eigenvectors for each of these.
For l_{1} = -1,
We have to solve:
= 2h_{1} + 2h_{2} ⇒ h_{1} = - h_{2}
The eigenvector for this case is,
= h_{2}= 1,
For l_{2} = 4,
We have to solve:
⇒ = -3h_{1} + 2h_{2} = 0 ⇒ h_{1} =( 2/3) h_{2}
The eigenvector for this case is,
= h_{2} = 3
So the general solution is after that,
Now, we have to get the constants. To do that we simply have to apply the initial conditions
All we required to do now is multiply the constants with and we after that get two equations that is one for each row which we can solve for the constants. It gives,
-c_{1} + 2c_{2} = 0;
c_{1} + 3c_{2} = -4;
By solving both equations we get:
c_{1} = -(8/5) and c_{2} = (-4/5)
The solution is subsequently,