61.28.22 problem 82

Internal problem ID [12503]
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 : 82
Date solved : Monday, March 31, 2025 at 05:36:53 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 46

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

\[ y = x^{-c +1} {\mathrm e}^{-\frac {x \left (a x +2 b \right )}{2}} \left (c_1 \int x^{-2+c} {\mathrm e}^{\frac {1}{2} a \,x^{2}+b x}d x +c_2 \right ) \]

✓ Mathematica. Time used: 0.393 (sec). Leaf size: 68

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

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

✗ 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)) + (2*a*x + b)*y(x) + (a*x**2 + b*x + c)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None