2.1043   ODE No. 1043

\[ y''(x)-x y'(x)+2 y(x)=0 \] Mathematica : cpu = 0.101548 (sec), leaf count = 69


\[\left \{\left \{y(x)\to \frac {1}{4} c_2 \left (\sqrt {2 \pi } x^2 \text {erfi}\left (\frac {x}{\sqrt {2}}\right )-\sqrt {2 \pi } \text {erfi}\left (\frac {x}{\sqrt {2}}\right )-2 e^{\frac {x^2}{2}} x\right )+c_1 \left (x^2-1\right )\right \}\right \}\] Maple : cpu = 0.218 (sec), leaf count = 39


\[y \relax (x ) = -2 \,{\mathrm e}^{\frac {x^{2}}{2}} c_{1} x +\left (x -1\right ) \left (1+x \right ) \left (\sqrt {\pi }\, \sqrt {2}\, \erfi \left (\frac {\sqrt {2}\, x}{2}\right ) c_{1}+c_{2}\right )\]

Hand solution

\begin {equation} y^{\prime \prime }-xy^{\prime }+2y=0\tag {1} \end {equation}

Second order with varying coefficient. Using power series, let \(y=\sum _{n=0}^{\infty }c_{n}x^{n}\), hence\begin {align*} y^{\prime } & =\sum _{n=0}^{\infty }nc_{n}x^{n-1}=\sum _{n=1}^{\infty }nc_{n}x^{n-1}=\sum _{n=0}^{\infty }\left (n+1\right ) c_{n+1}x^{n}\\ y^{\prime \prime } & =\sum _{n=0}^{\infty }n\left (n+1\right ) c_{n+1}x^{n-1}=\sum _{n=1}^{\infty }n\left (n+1\right ) c_{n+1}x^{n-1}=\sum _{n=0}^{\infty }\left (n+1\right ) \left (n+2\right ) c_{n+2}x^{n} \end {align*}

Substituting back in the original ODE gives\begin {align*} \sum _{n=0}^{\infty }\left (n+1\right ) \left (n+2\right ) c_{n+2}x^{n}-x\sum _{n=0}^{\infty }\left (n+1\right ) c_{n+1}x^{n}+2\sum _{n=0}^{\infty }c_{n}x^{n} & =0\\ \sum _{n=0}^{\infty }\left (n+1\right ) \left (n+2\right ) c_{n+2}x^{n}-\sum _{n=0}^{\infty }\left (n+1\right ) c_{n+1}x^{n+1}+\sum _{n=0}^{\infty }2c_{n}x^{n} & =0\\ \sum _{n=0}^{\infty }\left (n+1\right ) \left (n+2\right ) c_{n+2}x^{n}-\sum _{n=1}^{\infty }nc_{n}x^{n}+\sum _{n=0}^{\infty }2c_{n}x^{n} & =0 \end {align*}

For \(n=0\)\begin {align*} \left (n+1\right ) \left (n+2\right ) c_{n+2}+2c_{n} & =0\\ \relax (1) \relax (2) c_{2}+2c_{0} & =0\\ c_{2} & =-c_{0} \end {align*}

For \(n\geq 1\)\begin {align*} \left (n+1\right ) \left (n+2\right ) c_{n+2}-nc_{n}+2c_{n} & =0\\ c_{n+2} & =\frac {c_{n}\left (n-2\right ) }{\left (n+1\right ) \left ( n+2\right ) } \end {align*}

Hence for \(n=1\)\[ c_{3}=\frac {-c_{1}}{\relax (2) \relax (3) }\] For \(n=2\)\[ c_{4}=\frac {c_{2}\left (2-2\right ) }{\relax (3) \relax (4) }=0 \] For \(n=3\)\[ c_{5}=\frac {c_{3}}{\relax (4) \relax (5) }=\frac {-c_{1}}{\left ( 2\right ) \relax (3) \relax (4) \relax (5) }\] For \(n=4\) and since \(c_{4}=0\) then\[ c_{6}=\frac {c_{4}\left (n-2\right ) }{\left (n+1\right ) \left (n+2\right ) }=0 \] For \(n=5\)\[ c_{7}=\frac {3c_{5}}{\relax (6) \relax (7) }=-\frac {3c_{1}}{\relax (2) \relax (3) \relax (4) \relax (5) \relax (6) \relax (7) }\] For \(n=6\) and since \(c_{6}=0\) then\[ c_{8}=\frac {c_{6}\left (n-2\right ) }{\left (n+1\right ) \left (n+2\right ) }=0 \] For \(n=7\)\[ c_{9}=\frac {5c_{7}}{\relax (8) \relax (9) }=-\frac {\left ( 3\right ) \relax (5) c_{1}}{\relax (2) \relax (3) \left ( 4\right ) \relax (5) \relax (6) \relax (7) \left ( 8\right ) \relax (9) }\] And so on. Hence\begin {align*} y & =\sum _{n=0}^{\infty }c_{n}x^{n}\\ & =c_{0}+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+\cdots \\ & =c_{0}+c_{1}x-c_{0}x^{2}-\frac {c_{1}}{\relax (2) \relax (3) }x^{3}-\frac {c_{1}}{\relax (2) \relax (3) \relax (4) \relax (5) }x^{5}-\frac {3c_{1}}{\relax (2) \relax (3) \relax (4) \relax (5) \relax (6) \relax (7) }x^{7}-\frac {\relax (3) \relax (5) c_{1}}{\relax (2) \relax (3) \relax (4) \relax (5) \relax (6) \relax (7) \relax (8) \relax (9) }x^{9}-\cdots \\ & =c_{0}\left (1-x^{2}\right ) +c_{1}\left (x-\frac {1}{\relax (2) \relax (3) }x^{3}-\frac {1}{\relax (2) \relax (3) \left ( 4\right ) \relax (5) }x^{5}-\frac {3}{\relax (2) \left ( 3\right ) \relax (4) \relax (5) \relax (6) \left ( 7\right ) }x^{7}-\frac {\relax (3) \relax (5) }{\relax (2) \relax (3) \relax (4) \relax (5) \relax (6) \relax (7) \relax (8) \relax (9) }x^{9}\right ) \\ & =c_{0}\left (1-x^{2}\right ) +c_{1}\left (x-\frac {1}{3!}x^{3}-\frac {1}{5!}x^{5}-\frac {3}{7!}x^{7}-\frac {15}{9!}x^{9}-\cdots \right ) \end {align*}

Hence\[ y\relax (x) =c_{0}\left (1-x^{2}\right ) +c_{1}\left (x-\frac {1}{6}x^{3}-\frac {1}{120}x^{5}-\frac {1}{1680}x^{7}-\frac {1}{24\,192}x^{9}-\cdots \right ) \]

Verification