Internal problem ID [10863]
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: 29.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (-c +a \right ) x^{2}+b x +1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 43
dsolve(diff(y(x),x$2)+(a*x+b)*diff(y(x),x)+c*((a-c)*x^2+b*x+1)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} {\mathrm e}^{-\frac {x^{2} c}{2}}+c_{2} {\mathrm e}^{-\frac {x^{2} c}{2}} \operatorname {erf}\left (\frac {\left (a -2 c \right ) x +b}{\sqrt {2 a -4 c}}\right ) \]
✓ Solution by Mathematica
Time used: 0.135 (sec). Leaf size: 81
DSolve[y''[x]+(a*x+b)*y'[x]+c*((a-c)*x^2+b*x+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {1}{2} x (x (a-c)+2 b)} \left (c_1 \operatorname {HermiteH}\left (-1,\frac {b+(a-2 c) x}{\sqrt {2} \sqrt {a-2 c}}\right )+c_2 e^{\frac {(x (a-2 c)+b)^2}{2 (a-2 c)}}\right ) \]