60.3.209 problem 1224

Internal problem ID [11205]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1224
Date solved : Sunday, March 30, 2025 at 07:46:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 23

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

\[ y = c_1 \sin \left (\sqrt {a}\, \operatorname {arcsinh}\left (x \right )\right )+c_2 \cos \left (\sqrt {a}\, \operatorname {arcsinh}\left (x \right )\right ) \]

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 30

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

\[ y(x)\to c_1 \cos \left (\sqrt {a} \text {arcsinh}(x)\right )+c_2 \sin \left (\sqrt {a} \text {arcsinh}(x)\right ) \]

✗ Sympy

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

False