Inhomogeneous Systems (General Solutions)

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DaAzNJRiCh

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Please help me learn to do this step by step. I need within a couple hours if possible.

x'(t) = Matrix [ 1 3 ] x(t) + [1]
''''''''''''''''''''''''''''''''''''''''' [ 3 1 ]'''''''''''''''''[2]


a. Find the general solution using the method of undetermined coefficients

b. Find the general solution using the method variation of parameters

c. Find the general solution using that matrix [ 1 3 ] is diagonalizable
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''[ 3 1 ]

If you can only do one part at a time that's ok.
 
DaAzNJRiCh said:
Please help me learn to do this step by step. I need within a couple hours if possible.

x'(t) = Matrix [ 1 3 ] x(t) + [1]
''''''''''''''''''''''''''''''''''''''''' [ 3 1 ]'''''''''''''''''[2]


a. Find the general solution using the method of undetermined coefficients

b. Find the general solution using the method variation of parameters

c. Find the general solution using that matrix [ 1 3 ] is diagonalizable
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''[ 3 1 ]

If you can only do one part at a time that's ok.
Click to expand...

Please show us your work/thought - so that we know where to begin to help you.
 
The VP method, find the characteristic polynomial, It is \(\displaystyle {\lambda}^{2}-2{\lambda}-8=0\)

\(\displaystyle ({\lambda}+4)({\lambda}-2)=0\)

\(\displaystyle {\lambda}_{1}=-4, \;\ {\lambda}_{2}=2\)

Now, find the eigenvectors that correspond to \(\displaystyle {\lambda}_{1}, \;\ {\lambda}_{2}\)

Then you get \(\displaystyle X_{1}=\text{(eigenvector)}e^{-4t}, \;\ X_{2}=\text{(eigenvector)}e^{2t}\)

The entries in \(\displaystyle X_{1}\) form the first column in \(\displaystyle {\Phi}(t)\) and the entries in \(\displaystyle X_{2}\) form the second column in

\(\displaystyle {\Phi}(t)\)

Then set up:

\(\displaystyle X_{p}={\Phi}(t)\int{\Phi}^{-1}(t)F(t)\)

Where F(t) is the right of the equals in the original.

Then you should be able to put together your general solution.
 
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