Don't read this.
When you do you will realize why no one wanted to answer it.
The method is based on this tutorial.
Maxima:
Y:sum(a[n]*x^n,n,0,inf);
Y1:sum(n*a[n]*x^(n-1),n,1,inf);
Y2:sum(n*(n-1)*a[n]*x^(n-2),n,2,inf);
Y3:sum(n*(n-1)*(n-2)*a[n]*x^(n-3),n,3,inf);
E1 : c1*Y3 + c2*Y2 + (c3*x + c4 + c5*x^2)*Y1 + Y*(c6*x + c7*x^3);
E2 : expand(E1);
E3 : intosum(E2);
E4 : factorsum(E3);
E5 : changevar(part(E4,1),k-n-3,k,n)+changevar(part(E4,2)+part(E4,3),k-n-1,k,n)
+changevar(part(E4,4),k-n,k,n)+changevar(part(E4,5),k-n+1,k,n)
+changevar(part(E4,6),k-n+2,k,n)+changevar(part(E4,7),k-n+3,k,n);
E6 : intosum(E5);
E7 : sumcontract(E6);
E8 : factorsum(E7);
Coeffs1 : REC:coeff(part(E8,1,1),x,k);
$$Y = \sum_{n=0}^{\infty }{a_{n}\,x^{n}} \tag{1}$$
$$Y1 = \frac{dy}{dx} = \sum_{n=1}^{\infty }{n\,a_{n}\,x^{n-1}} \tag{2}$$
$$Y2 = \frac{d^2y}{dx^2} = \sum_{n=2}^{\infty }{\left(n-1\right)\,n\,a_{n}\,x^{n-2}} \tag{3}$$
$$Y3 = \frac{d^3y}{dx^3} =\sum_{n=3}^{\infty }{\left(n-2\right)\,\left(n-1\right)\,n\,a_{n}\,
x^{n-3}} \tag{4}$$
The equation $c_1f'''(x)+c_2 f''(x) + (c_3x + c_4 +c_5x^2)f'(x) + f(x) ( c_6x + c_7 x^3) =0$
$$E1 = \left({\it c_7}\,x^3+{\it c_6}\,x\right)\,\sum_{n=0}^{\infty }{a_{n
}\,x^{n}}+\left({\it c_5}\,x^2+{\it c_3}\,x+{\it c_4}\right)\,\sum_{
n=1}^{\infty }{n\,a_{n}\,x^{n-1}}+{\it c_2}\,\sum_{n=2}^{\infty }{
\left(n-1\right)\,n\,a_{n}\,x^{n-2}}+{\it c_1}\,\sum_{n=3}^{\infty
}{\left(n-2\right)\,\left(n-1\right)\,n\,a_{n}\,x^{n-3}} \tag{5}$$
Expansion of terms:
$$E2 = {\it c_7}\,x^3\,\sum_{n=0}^{\infty }{a_{n}\,x^{n}}+{\it c_6}\,x\,
\sum_{n=0}^{\infty }{a_{n}\,x^{n}}+{\it c_5}\,x^2\,\sum_{n=1}^{
\infty }{n\,a_{n}\,x^{n-1}}+{\it c_3}\,x\,\sum_{n=1}^{\infty }{n\,a
_{n}\,x^{n-1}}+{\it c_4}\,\sum_{n=1}^{\infty }{n\,a_{n}\,x^{n-1}}+
{\it c_2}\,\sum_{n=2}^{\infty }{\left(n^2\,a_{n}\,x^{n-2}-n\,a_{n}\,
x^{n-2}\right)}+{\it c_1}\,\sum_{n=3}^{\infty }{\left(n^3\,a_{n}\,x
^{n-3}-3\,n^2\,a_{n}\,x^{n-3}+2\,n\,a_{n}\,x^{n-3}\right)} \tag{6}$$
Move the $x$ multipliers into the sums.
$$E3 = \sum_{n=0}^{\infty }{{\it c_7}\,a_{n}\,x^{n+3}}+\sum_{n=1}^{\infty
}{{\it c_5}\,n\,a_{n}\,x^{n+1}}+\sum_{n=0}^{\infty }{{\it c_6}\,a_{n
}\,x^{n+1}}+\sum_{n=1}^{\infty }{{\it c_3}\,n\,a_{n}\,x^{n}}+\sum_{n=
1}^{\infty }{{\it c_4}\,n\,a_{n}\,x^{n-1}}+\sum_{n=2}^{\infty }{
{\it c_2}\,\left(n^2\,a_{n}\,x^{n-2}-n\,a_{n}\,x^{n-2}\right)}+
\sum_{n=3}^{\infty }{{\it c_1}\,\left(n^3\,a_{n}\,x^{n-3}-3\,n^2\,a
_{n}\,x^{n-3}+2\,n\,a_{n}\,x^{n-3}\right)} \tag{7}$$
Collect factors:
$$E4 = {\it c_7}\,\sum_{n=0}^{\infty }{a_{n}\,x^{n+3}}+{\it c_5}\,\sum_{n=
1}^{\infty }{n\,a_{n}\,x^{n+1}}+{\it c_6}\,\sum_{n=0}^{\infty }{a_{n
}\,x^{n+1}}+{\it c_3}\,\sum_{n=1}^{\infty }{n\,a_{n}\,x^{n}}+
{\it c_4}\,\sum_{n=1}^{\infty }{n\,a_{n}\,x^{n-1}}+{\it c_2}\,\sum_{
n=2}^{\infty }{\left(n-1\right)\,n\,a_{n}\,x^{n-2}}+{\it c_1}\,
\sum_{n=3}^{\infty }{n\,\left(n^2-3\,n+2\right)\,a_{n}\,x^{n-3}} \tag{8}$$
Set all powers to $x^k$:
$$E5 = {\it c_1}\,\sum_{k=0}^{\infty }{\left(k^3+6\,k^2+11\,k+6\right)\,a
_{k+3}\,x^{k}}+{\it c_2}\,\sum_{k=0}^{\infty }{\left(k^2+3\,k+2
\right)\,a_{k+2}\,x^{k}}+{\it c_4}\,\sum_{k=0}^{\infty }{\left(k+1
\right)\,a_{k+1}\,x^{k}}+{\it c_3}\,\sum_{k=1}^{\infty }{k\,a_{k}\,x
^{k}}+{\it c_5}\,\sum_{k=2}^{\infty }{\left(k-1\right)\,a_{k-1}\,x^{
k}}+{\it c_6}\,\sum_{k=1}^{\infty }{a_{k-1}\,x^{k}}+{\it c_7}\,
\sum_{k=3}^{\infty }{a_{k-3}\,x^{k}} \tag{9}$$
Move the $c_k$ multipliers into the sums:
$$E6 = \sum_{k=0}^{\infty }{{\it c_1}\,\left(k^3+6\,k^2+11\,k+6\right)\,a
_{k+3}\,x^{k}}+\sum_{k=0}^{\infty }{{\it c_2}\,\left(k^2+3\,k+2
\right)\,a_{k+2}\,x^{k}}+\sum_{k=0}^{\infty }{{\it c_4}\,\left(k+1
\right)\,a_{k+1}\,x^{k}}+\sum_{k=1}^{\infty }{{\it c_3}\,k\,a_{k}\,x
^{k}}+\sum_{k=2}^{\infty }{{\it c_5}\,\left(k-1\right)\,a_{k-1}\,x^{
k}}+\sum_{k=1}^{\infty }{{\it c_6}\,a_{k-1}\,x^{k}}+\sum_{k=3}^{
\infty }{{\it c_7}\,a_{k-3}\,x^{k}} \tag{10}$$
Collect like terms:
$$E7 = \sum_{k=3}^{\infty }{\left({\it c_1}\,\left(k^3+6\,k^2+11\,k+6
\right)\,a_{k+3}\,x^{k}+{\it c_2}\,\left(k^2+3\,k+2\right)\,a_{k+2}
\,x^{k}+{\it c_4}\,\left(k+1\right)\,a_{k+1}\,x^{k}+{\it c_3}\,k\,a
_{k}\,x^{k}+{\it c_5}\,\left(k-1\right)\,a_{k-1}\,x^{k}+{\it c_6}\,a
_{k-1}\,x^{k}+{\it c_7}\,a_{k-3}\,x^{k}\right)}+a_{1}\,{\it c_6}\,x^
2+a_{1}\,{\it c_5}\,x^2+3\,a_{3}\,{\it c_4}\,x^2+2\,a_{2}\,{\it c_3}
\,x^2+12\,a_{4}\,{\it c_2}\,x^2+60\,a_{5}\,{\it c_1}\,x^2+a_{0}\,
{\it c_6}\,x+2\,a_{2}\,{\it c_4}\,x+a_{1}\,{\it c_3}\,x+6\,a_{3}\,
{\it c_2}\,x+24\,a_{4}\,{\it c_1}\,x+a_{1}\,{\it c_4}+2\,a_{2}\,
{\it c_2}+6\,a_{3}\,{\it c_1} \tag{11}$$
Collect coefficients of $x^k$.
$E8$ collects all the $x^k$ coefficients inside the sum and all the stray terms outside of it.
Note: $k$ starts from $3$.
$$E8 : \sum_{k=3}^{\infty }{\left(\left({\it c_1}\,k^3+6\,{\it c_1}\,k^2+
11\,{\it c_1}\,k+6\,{\it c_1}\right)\,a_{k+3}+\left({\it c_2}\,k^2+3
\,{\it c_2}\,k+2\,{\it c_2}\right)\,a_{k+2}+\left({\it c_4}\,k+
{\it c_4}\right)\,a_{k+1}+{\it c_3}\,k\,a_{k}+\left({\it c_5}\,k+
{\it c_6}-{\it c_5}\right)\,a_{k-1}+{\it c_7}\,a_{k-3}\right)\,x^{k}
}+a_{1}\,{\it c_6}\,x^2+a_{1}\,{\it c_5}\,x^2+3\,a_{3}\,{\it c_4}\,x
^2+2\,a_{2}\,{\it c_3}\,x^2+12\,a_{4}\,{\it c_2}\,x^2+60\,a_{5}\,
{\it c_1}\,x^2+6\,{\it c_1}\,\left(4\,a_{4}\,x+a_{3}\right)+2\,
{\it c_2}\,\left(3\,a_{3}\,x+a_{2}\right)+a_{0}\,{\it c_6}\,x+2\,a_{
2}\,{\it c_4}\,x+a_{1}\,{\it c_3}\,x+a_{1}\,{\it c_4} = 0 \tag{12}$$
coefficients of $x^k$:
$$Coeffs1 : \left({\it c_1}\,k^3+6\,{\it c_1}\,k^2+11\,{\it c_1}\,k+6\,
{\it c_1}\right)\,a_{k+3}+\left({\it c_2}\,k^2+3\,{\it c_2}\,k+2\,
{\it c_2}\right)\,a_{k+2}+\left({\it c_4}\,k+{\it c_4}\right)\,a_{k+
1}+{\it c_3}\,k\,a_{k}+\left({\it c_5}\,k+{\it c_6}-{\it c_5}\right)
\,a_{k-1}+{\it c_7}\,a_{k-3} \tag{13} = 0$$
From $E8$ collect all the coefficients of $x^k$ for $k=0..2$:
$$ 6 c_1 a_{3} +2 c_2 a_{2} +a_1 c_4 = 0 $$
$$ 24c_1a_4 + 3a_2 + a_0c_6 + 2a_2c_4 + a_1c_3 = 0 $$
$$ a_1c_6 + a_1c_5 + 3a_2c_4 + 2a_2c_3 + 12a_4c_2 + 60a_5c_1 = 0$$
$a_0$..$a_5$ need to be chosen to satisfy these equations. There could be arbitrarily many solutions.
The other coefficients $a_6$... can be calculated from $Coeffs1$ equation $(13)$
At this point we need to be reminded that $\displaystyle Y = \sum_{n=0}^{\infty }{a_{n}\,x^{n}} $
