55.29.17 problem 77

Internal problem ID [13850]
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 : 77
Date solved : Thursday, October 02, 2025 at 08:07:46 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 112

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

\[ y = \frac {c_2 \,a^{3}+\int \left (\left (\left (b^{3}+3 b^{2}+2 b \right ) \Gamma \left (b , -a x \right )-\Gamma \left (b +3\right )\right ) c^{2} {\mathrm e}^{-a x} \left (-a x \right )^{-b}+x^{-b} {\mathrm e}^{-a x} c_1 \,a^{3}-c \left (\left (-b^{2}+\left (a x -3\right ) b -a^{2} x^{2}+2 a x -2\right ) c +x \,a^{3}\right )\right )d x}{a^{3}} \]

✓ Mathematica. Time used: 61.169 (sec). Leaf size: 92

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

\begin{align*} y(x)&\to \int _1^xe^{-a K[1]} K[1]^{-b} \left (\frac {c \left (-\left ((b+1) \Gamma (b+1,-a K[1]) a^2\right )+\Gamma (b+2,-a K[1]) a^2+c \Gamma (b+3,-a K[1])\right ) K[1]^b (-a K[1])^{-b}}{a^3}+c_1\right )dK[1]+c_2 \end{align*}

✗ Sympy

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

Timed Out