Internal problem ID [10848]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-2 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 14.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2}+a x +1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 40
dsolve(diff(y(x),x$2)+a*diff(y(x),x)+b*(-b*x^2+a*x+1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \operatorname {erf}\left (x \sqrt {-b}+\frac {a}{2 \sqrt {-b}}\right ) {\mathrm e}^{-\frac {b \,x^{2}}{2}} c_{1} +c_{2} {\mathrm e}^{-\frac {b \,x^{2}}{2}} \]
✓ Solution by Mathematica
Time used: 0.28 (sec). Leaf size: 67
DSolve[y''[x]+a*y'[x]+b*(-b*x^2+a*x+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} e^{-\frac {b x^2}{2}} \left (\frac {\sqrt {\pi } c_2 e^{-\frac {a^2}{4 b}} \text {erfi}\left (\frac {2 b x-a}{2 \sqrt {b}}\right )}{\sqrt {b}}+2 c_1\right ) \]