60.3.215 problem 1230

Internal problem ID [11211]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1230
Date solved : Sunday, March 30, 2025 at 07:46:15 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Maple. Time used: 0.040 (sec). Leaf size: 36

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

\[ y = c_1 \left (x^{2}+1\right )^{-\frac {a}{2}+1}+c_2 x \operatorname {hypergeom}\left (\left [1, \frac {a}{2}-\frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right ) \]

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 68

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

\[ y(x)\to \left (x^2+1\right )^{\frac {1}{2}-\frac {a}{4}} \left (c_1 P_{\frac {a-4}{2}}^{\frac {a-2}{2}}(i x)+c_2 Q_{\frac {a-4}{2}}^{\frac {a-2}{2}}(i x)\right ) \]

✗ Sympy

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

False