Introduction: In this question, we find power series solution about x0 =0 example 1 of an Ordinary differential equation. Step by step solution is given below
Question: Find power series solution about x0 =0 of differential equation
\[\left(1+x^2\right)y^{\prime\prime}-xy^\prime+y=e^{-x}\]
Solution:
given differential equation
\[\left(1+x^2\right)y^{\prime\prime}-xy^\prime+y=e^{-x} \] (1)
Let power series solution of the differential eq.(1) about x= 0 is given by
\[y=\sum_{n=0}^{\infty}{a_nx^n} \] (2)
differentiating it
\[\Rightarrow y^\prime=\sum_{n=1}^{\infty}{na_nx^{n-1}, {y^\prime}^\prime= \sum_{n=2}^{\infty}{n(n-1)a_nx^{n-2}}}\]
(3)
Substituting these expressions into eqn (1)
\[\sum_{n=2}^{\infty}{n\left(n-1\right)a_nx^{n-2}}+\sum_{n=2}^{\infty}{n\left(n-1\right)a_nx^n}-\sum_{n=1}^{\infty}{na_nx^n+\sum_{n=0}^{\infty}{a_nx^n}=e^{-x}}\]
we can write it as follows
\[\Rightarrow\sum_{n=0}^{\infty}{(n+1)\left(n+2\right)a_{n+2}x^n}+\sum_{n=2}^{\infty}{n\left(n-1\right)a_nx^n-\sum_{n=1}^{\infty}{na_nx^n+\sum_{n=0}^{\infty}{a_nx^n}=e^{-x}}}\]
\[2a_2+6a_3x-a_1x+a_0+a_1x+\sum_{n=2}^{\infty}[(n+1)(n+2)a_{n+2}+n(n-1)a_n-na_n +a_n] x^n=1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+\ldots\]
Comparing coefficients both sides
\[{2a}_2+a_0=1\Rightarrow a_2=\frac{\left(1-a_0\right)}{2}, 6a_3=-1\Rightarrow a_3=-\frac{1}{6}\]
and recurrence relation
\[\left(n+1\right)\left(n+2\right)a_{n+2}+n\left(n-1\right)a_n-{na}_{n\ }+a_n=\frac{\left(-1\right)^n}{n!}, n\geq 2\]
or
\[\Rightarrow a_{n+2}=\frac{\left(-1\right)^n}{n!\left(n+1\right)\left(n+2\right)}-\frac{n^2-2n+1}{\left(n+1\right)\left(n+2\right)}a_n , n\geq 2\]
put n=2
\[ \Rightarrow a_4=\frac{1}{2.3.4}-\frac{1}{3.4}\frac{\left(1-a_0\right)}{2}=\frac{1}{2.3.4}a_0\]
put n=3
\[\Rightarrow a_5=-\frac{1}{6.4.5}-\frac{4}{4.5}a_3=\frac{3}{6.4.5}\]
put n=4
\[\Rightarrow a_6=\frac{1}{24.5.6}-\frac{9}{5.6}\frac{1}{2.3.4}a_0\]
now substituting in to series
\[y\left(x\right)=a_0+a_1x+\frac{\left(1-a_0\right)}{2}x^2-\frac{1}{6}x^3+\frac{1}{2.3.4}a_0x^4+\frac{3}{6.4.5}x^5+\left(\frac{1}{24.5.6}-\frac{9}{5.6}\frac{1}{2.3.4}a_0\right)x^6+\ldots.\]
\[y\left(x\right)=a_0\left( 1-\frac{1}{2}x^2+\frac{1}{2.3.4}x^4-\frac{9}{5.6}\frac{1}{2.3.4}x^6+…..\right)+a_1x+\frac{x^2}{2}-\frac{x^3}{6}+\frac{1}{24.5.6}x^6+….\]
Which is the desired power series solution about x0=0 of the given differential equation.