61.30.26 problem 174

Internal problem ID [12595]
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 : 174
Date solved : Monday, March 31, 2025 at 05:52:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 43

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

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

✓ Mathematica. Time used: 0.155 (sec). Leaf size: 63

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

\[ y(x)\to c_1 \cosh \left (m \log \left (\sqrt {2 a x+b+x^2}-a-x\right )\right )-i c_2 \sinh \left (m \log \left (\sqrt {2 a x+b+x^2}-a-x\right )\right ) \]

✗ Sympy

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

False