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Asked Oct 10, 2019
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Expert Answer

Step 1

First find the solution of the second order linear, homogeneous ODE   y’’= y’ + y

 

Fibonacci numbers recursively defined by

 

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n-2 n-1 so, F FF1 F FF 2 F FF3 Hence F n1 for n > 1

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Step 2

Assume that the solution is of the form:

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y ()-Σ. F = F+Fx+2 x2+ 2! t... 3! 1 =x+ 2! 2 3 1-1 -x + x* + 3! 4! n! y(x)=cx ..1) C. n!

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Step 3

 

Differentiate the power ...

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F R+Fx+ 2 ... y1+x+x or y Σ , π" .-(2) nc y"1+2x y- Σn(n-1) c,x" .(3) or

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