Question

Find the terms of the power series solution to the DE (x^2-3x+1) y''+(2x-1)y'-2y=-16x^3+12x^2-8 with initial conditions y(0)=-1, y'(0)=10

Answer 1

Here are the first few terms of your solution
\[y(x)=-1+10 x+x^4+\frac{6 x^5}{5}+2 x^6+\frac{132 x^7}{35}+\\ \frac{527 x^8}{70}+\frac{4906 x^9}{315}+\frac{6947
x^{10}}{210}+O\left(x^{11}\right)\]

Answer 2

I just gave it to you to check your answer which should be in symbolic form

Answer 3

I once wrote a Mathematica Program to solve such a question. The method of series solutions is really nothing else than data processing and should be done by a computer.
It is easy but requires a lot of time and energy. You should be able to do it. I can follow your steps.

Answer 4

\[\Rightarrow \sum_{n=2}^{\infty}n(n-1)a_nx^n-3\sum_{n=2}^{\infty}n(n-1)a_{n}x^{n-1}+\sum_{n=2}^{\infty}n(n-1)a_{n}x^{n-2}\]
\[+2\sum_{n=1}^{\infty}na_nx^n-\sum_{n=1}^{\infty}na_nx^{n-1}-2\sum_{n=0}^{\infty}a_nx^n=16x^3+12x^2-8\]



\[\Rightarrow \sum_{n=2}^{\infty}n(n-1)a_nx^n-3\sum_{n=1}^{\infty}n(n+1)a_{n+1}x^n+\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n\]
\[+2\sum_{n=1}^{\infty}na_nx^n-\sum_{n=0}^{\infty}(n+1)a_nx^n-2\sum_{n=0}^{\infty}a_nx^n=16x^3+12x^2-8\]


\[\Rightarrow\sum_{n=2}^{\infty}n(n-1)a_nx^n-6a_2x-3\sum_{n=2}^{\infty}n(n+1)a_{n+1}x^n+2a_2+6a_3x+\]
\[\sum_{n=2}^{\infty}(n+2)(n+1)a_{n+2}x^n
+2a_1x+2\sum_{n=2}^{\infty}na_nx^n-a_0-2a_1x-\sum_{n=2}^{\infty}(n+1)a_nx^n-\]
\[2a_0-2a_1x-2\sum_{n=2}^{\infty}a_nx^n=16x^3+12x^2-8\]

Answer 5

@eliassaab Would you check that for me and see whether i did it right

Answer 6

That does not give \( a_0=-1\) and \(a_1=10\)

Answer 7

The solution should find \(a_0=-1\) and \(a_1=10\) and a recurrence formula to find all the \(a_n\) for \( n\ge 2\)