61.28.20 problem 80

Internal problem ID [12501]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 80
Date solved : Monday, March 31, 2025 at 05:36:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.130 (sec). Leaf size: 34

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

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

✓ Mathematica. Time used: 0.774 (sec). Leaf size: 49

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

\[ y(x)\to x e^{c x} \left (c_2 \int _1^x\frac {e^{-\frac {1}{2} K[1] (2 b+4 c+a K[1])}}{K[1]^2}dK[1]+c_1\right ) \]

✗ Sympy

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

False