Internal problem ID [7472]
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]]
\[ \boxed {y^{\prime \prime }-2 b x y^{\prime }+b^{2} x^{2} y=x} \]
✓ Solution by Maple
Time used: 0.015 (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)
\[ y \left (x \right ) = {\mathrm e}^{\frac {x \left (x b +2 \sqrt {-b}\right )}{2}} c_{2} +{\mathrm e}^{\frac {x \left (x b -2 \sqrt {-b}\right )}{2}} c_{1} +\frac {\sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{-\frac {1}{2}+\frac {b \,x^{2}}{2}-x \sqrt {-b}} \left (-\operatorname {erf}\left (\frac {\sqrt {2}\, \left (x b +\sqrt {-b}\right )}{2 \sqrt {b}}\right ) {\mathrm e}^{2 x \sqrt {-b}}+\operatorname {erf}\left (\frac {\sqrt {2}\, \left (-x b +\sqrt {-b}\right )}{2 \sqrt {b}}\right )\right )}{4 b^{\frac {3}{2}}} \]
✓ Solution by Mathematica
Time used: 0.427 (sec). Leaf size: 139
DSolve[y''[x]-2*b*x*y'[x]+b^2*x^2*y[x]==x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{\frac {1}{2} \left (\sqrt {b} x-i\right )^2} \left (-\sqrt {2 \pi } e^{2 i \sqrt {b} x} \text {erf}\left (\frac {\sqrt {b} x+i}{\sqrt {2}}\right )+i \sqrt {2 \pi } \text {erfi}\left (\frac {1+i \sqrt {b} x}{\sqrt {2}}\right )+2 \sqrt {e} b \left (2 \sqrt {b} c_1-i c_2 e^{2 i \sqrt {b} x}\right )\right )}{4 b^{3/2}} \]