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Q: Math question ( Answered ,   0 Comments ) Question
 Subject: Math question Category: Science > Math Asked by: edwardwen85-ga List Price: \$2.50 Posted: 05 Dec 2004 14:34 PST Expires: 04 Jan 2005 14:34 PST Question ID: 438489
 ```write the third order differential equation y'''(t)-2y''(t)+3y'(t)+4y(t)=0 as an equivalent linear system of first order differential equations in matrix form x'(t)=Ax(t)``` Subject: Re: Math question Answered By: calebu2-ga on 05 Dec 2004 15:12 PST Rated: ```Edwardwen85, I am going to answer this question from first principles. As a result, I will include information that one would not normally include if simply answering a textbook question - rather I will try and explain a little of what is going on. If this question is for a course, I would encourage you to reformat the answer to fit the style of the course (different people teach this material in different ways). The first step in rewriting this equation is to create a mapping between the various derivatives of y and the components of the vector x. Write x1 = y'', x2 = y', x3 = y. Then x1' = y''', x2' = y'' = x1, x3' = y' = x2 We can incorporate this information into a matrix as follows : (x1') (? ? ?) (x1) (x2') = (1 0 0) (x2) (x3') (0 1 0) (x3) All that remains is to represent the actual differential equation, y''' - 2y'' + 3y' + 4y = 0 in matrix form. Rearranging and substituting x1', x1, x2, x3 for y''', y'', y' and y respectively we get : x1' = 2*x1 - 3*x2 - 4*x3. Hence the top line of the matrix becomes 2, -3, -4. So the matrix form of the differential equation is : (x1') (2 -3 -4) (x1) (x2') = (1 0 0) (x2) (x3') (0 1 0) (x3) This question can be split into two key steps : 1) Write down a relationship between the derivatives of y and translate them into a simple relationship between elements of the vector x. and 2) Rewrite the original differential equation in this form. (Step 3 is to write out the answer to parts 1 and 2 in matrix form). Good luck. calebu2-ga```
 edwardwen85-ga rated this answer: and gave an additional tip of: \$1.00 `great explanation`  