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Here is some verbatim
problem Transform the following problem or system to set of first order ODE
solution Since this is second order ODE, we need two state variables, say
Let , hence
Hence the two first order ODE’s are (now coupled)
The matrix form of the above is
problem This problem was solved in textbook using matrix exponential. Here is solved using the fundamental matrix only. Use the method of variation of parameters to solve .
Solution
The homogeneous solution was found in the book as
Following scalar case, the guess would be but since is in the homogeneous, we have to adjust to be . Notice we had to add , else it will not work if we just guessed based on what we would do in scalar case, we will find we get . This seems to be a trial and error stage and one just have to try to find out. This is why undermined coefficients for systems is not as easy to use as with scalar case. Hence
Now we plug-in this back into the ODE and solve for . But an easier method is to use Variation of parameters. The fundamental matrix is
And
Hence using
The integral of and (using integration by parts) hence the above simplifies to
Hence the complete solution is
To find the constants, we apply initial conditions. At
Hence or and , hence . Therefore the solution becomes