61.30.29 problem 177

Internal problem ID [12598]
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 : 177
Date solved : Monday, March 31, 2025 at 05:53:13 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 79

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

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

✓ Mathematica. Time used: 0.213 (sec). Leaf size: 93

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

\[ y(x)\to c_1 \cos \left (\frac {\sqrt {d} \log \left (-\sqrt {a} \sqrt {a x^2+2 b x+c}+a x+b\right )}{\sqrt {a}}\right )-c_2 \sin \left (\frac {\sqrt {d} \log \left (-\sqrt {a} \sqrt {a x^2+2 b x+c}+a x+b\right )}{\sqrt {a}}\right ) \]

✗ 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) + (a*x + 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