61.27.19 problem 29

Internal problem ID [12450]
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
Problem number : 29
Date solved : Monday, March 31, 2025 at 05:34:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b x +1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 36

ode:=diff(diff(y(x),x),x)+(a*x+b)*diff(y(x),x)+c*((a-c)*x^2+b*x+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {x^{2} c}{2}} \left (c_2 \,\operatorname {erf}\left (\frac {\left (-2 c +a \right ) x +b}{\sqrt {2 a -4 c}}\right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.082 (sec). Leaf size: 81

ode=D[y[x],{x,2}]+(a*x+b)*D[y[x],x]+c*((a-c)*x^2+b*x+1)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-\frac {1}{2} x (x (a-c)+2 b)} \left (c_1 \operatorname {HermiteH}\left (-1,\frac {b+(a-2 c) x}{\sqrt {2} \sqrt {a-2 c}}\right )+c_2 e^{\frac {(x (a-2 c)+b)^2}{2 (a-2 c)}}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(c*(b*x + x**2*(a - c) + 1)*y(x) + (a*x + b)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False