55.31.27 problem 175

Internal problem ID [13948]
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 : 175
Date solved : Thursday, October 02, 2025 at 08:21:10 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Maple. Time used: 0.323 (sec). Leaf size: 1400

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

\[ \text {Expression too large to display} \]

✓ Mathematica. Time used: 6.507 (sec). Leaf size: 164

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

\begin{align*} y(x)&\to (x (a x+b)+c)^{1-\frac {d}{2 a}} \exp \left (\frac {(b d-2 a k) \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{a \sqrt {4 a c-b^2}}\right ) \left (c_2 \int _1^x\exp \left (\frac {(d-4 a) \log (c+K[1] (b+a K[1]))-\frac {2 (b d-2 a k) \arctan \left (\frac {b+2 a K[1]}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}}{2 a}\right )dK[1]+c_1\right ) \end{align*}

✗ Sympy

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

False