Science:Math Exam Resources/Courses/MATH215/December 2013/Question 05 (b)
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Question 05 (b) 

Now find the general solution of the nonhomogenous system (with matrix as above): Hint: to find a particular solution, you may use either the method of variation of parameters, or the method of undetermined coefficients; for the latter, try 
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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

If you use undetermined coefficients like in the hint, substitute into the equation, being careful with all your terms. Be sure to include the nonhomogeneous term in your resulting equation. You need to solve for a, b, c, and d. 
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Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. We need a particular solution. To obtain it, we will use the method of undetermined coefficients. If then and Therefore if we get
In comparing the coefficients of from row 1, the coefficients of from row 1, the coefficients of from row 2, and the coefficients of from row 2, the following equations must hold: We can turn this into a matrix equation and rowreduce: Subtracting row 2 from row 4 and row 1 from row 2 yields: We next multiply the third row by 3 and then add the second row twice: Next, divide the third row by 5 and then add the result to the second row, and subtract it from the first row: In our last step we divide the second row by 3 and then subtract the result from the first row: We see that d is a free parameter, and since we only need one particular we choose d=0 for simplicity. Then we read off a = 10, b  d = 5, c = 10. Putting this together we find a particular solution . The general solution is the homogeneous solution (which we found in part (a)) plus a particular solution: . 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. This problem can also be done using variation of parameters. Given the homogeneous solution , we form a fundamental matrix using independent solutions of the homogeneous problem as columns. Observe that . For the nonhomogeneous equation, we will try to find a solution of the form where With this solution form, the equation reads:
We can now work on solving for (note the primed in ).
Integrating gives
We finally have a general solution . This is in agreement with the first solution. Observe that based on the numbers in that solution, a=c=10, and bd=5. 