55.31.30 problem 178

Internal problem ID [13951]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-5
Problem number : 178
Date solved : Thursday, October 02, 2025 at 08:21:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (a \,x^{2}+2 b x +c \right ) y^{\prime \prime }+3 \left (a x +b \right ) y^{\prime }+d y&=0 \end{align*}

✓ Maple. Time used: 0.161 (sec). Leaf size: 88

ode:=(a*x^2+2*b*x+c)*diff(diff(y(x),x),x)+3*(a*x+b)*diff(y(x),x)+d*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.094 (sec). Leaf size: 152

ode=(a*x^2+2*b*x+c)*D[y[x],{x,2}]+3*(a*x+b)*D[y[x],x]+d*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_1 P_{\frac {\sqrt {a-d}}{\sqrt {a}}-\frac {1}{2}}^{\frac {1}{2}}\left (\frac {\sqrt {-b^2-a c} (b+a x)}{a \sqrt {c^2-\frac {b^4}{a^2}}}\right )+c_2 Q_{\frac {\sqrt {a-d}}{\sqrt {a}}-\frac {1}{2}}^{\frac {1}{2}}\left (\frac {\sqrt {-b^2-a c} (b+a x)}{a \sqrt {c^2-\frac {b^4}{a^2}}}\right )}{\sqrt [4]{x (a x+2 b)+c}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
y = Function("y") 
ode = Eq(d*y(x) + (3*a*x + 3*b)*Derivative(y(x), x) + (a*x**2 + 2*b*x + c)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False