Internal problem ID [6716]
Book: Second order enumerated odes
Section: section 2
Problem number: 31.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-2 b x y^{\prime }+b^{2} x^{2} y-x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 116
dsolve(diff(y(x),x$2)-2*b*x*diff(y(x),x)+b^2*x^2*y(x)=x,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{\frac {x \left (b x +2 \sqrt {-b}\right )}{2}} c_{2}+{\mathrm e}^{\frac {x \left (b x -2 \sqrt {-b}\right )}{2}} c_{1}+\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {1}{2}+\frac {b \,x^{2}}{2}-x \sqrt {-b}} \sqrt {2}\, \left (-\erf \left (\frac {\sqrt {2}\, \left (b x +\sqrt {-b}\right )}{2 \sqrt {b}}\right ) {\mathrm e}^{2 x \sqrt {-b}}+\erf \left (\frac {\sqrt {2}\, \left (-b x +\sqrt {-b}\right )}{2 \sqrt {b}}\right )\right )}{4 b^{\frac {3}{2}}} \]
✓ Solution by Mathematica
Time used: 0.179 (sec). Leaf size: 116
DSolve[y''[x]-2*b*x*y'[x]+b^2*x^2*y[x]==x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{\frac {b x^2}{2}-i \sqrt {b} x} \left (4 b^{3/2} c_1-2 i b c_2 e^{2 i \sqrt {b} x}\right )-2 i \sqrt {2} \left (F\left (\frac {1-i \sqrt {b} x}{\sqrt {2}}\right )-F\left (\frac {i \sqrt {b} x+1}{\sqrt {2}}\right )\right )}{4 b^{3/2}} \\ \end{align*}