Solving system of differential equations : Wolfram Alpha vs theoremI am burning my brain finding the most correct way to solve a system of differential equations. Here is an example :{x′=5x−2yy′=−x+6y&nbsp;Let's&nbsp;Y(t)=x(t)y(t). I use a matrix&nbsp;A=5-2-16, calculate its eigen values&nbsp;λ1=7,λ2=4&nbsp;and eigen vectors&nbsp;v1=-11,v2=21.Now 2 options:- I can use a theorem and find the solution is&nbsp;Y(t)=c1eλ1tu1+c2eλ2tu2- I can continue using linear algebra, calculate P, D and&nbsp;P−1&nbsp;such that&nbsp;A=PDP−1&nbsp;where D is a diagonal matrix.&nbsp;Y′=AY&nbsp;so&nbsp;Y=CetA=CPe7t00e4tP-1I don't find same results but both are OK (the second is used by Wolfram). Did I misunderstand something? Which one is the best option?

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Solving system of differential equations : Wolfram Alpha vs theoremI am burning my brain finding the most correct way to solve a system of differential equations. Here is an example :{x′=5x−2yy′=−x+6y&amp;nbsp;Let&#039;s&amp;nbsp;Y(t)=x(t)y(t). I use a matrix&amp;nbsp;A=5-2-16, calculate its eigen values&amp;nbsp;λ1=7,λ2=4&amp;nbsp;and eigen vectors&amp;nbsp;v1=-11,v2=21.Now 2 options:- I can use a theorem and find the solution is&amp;nbsp;Y(t)=c1eλ1tu1+c2eλ2tu2- I can continue using linear algebra, calculate P, D and&amp;nbsp;P−1&amp;nbsp;such that&amp;nbsp;A=PDP−1&amp;nbsp;where D is a diagonal matrix.&amp;nbsp;Y′=AY&amp;nbsp;so&amp;nbsp;Y=CetA=CPe7t00e4tP-1I don&#039;t find same results but both are OK (the second is used by Wolfram). Did I misunderstand something? Which one is the best option?

Makenzie Hart · Accepted Answer

Explanation:If you were to write that last formula correctly asY(t)=etAC=Pe7t00e4tP-1C,&amp;nbsp;&amp;nbsp;&amp;nbsp;C=Y(0),then you get the correspondence with the first formula withP−1C=(c1c2),and the columns of P being the eigenvectors of A.

Solving system of differential equations : Wolfram Alpha

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