60.3.211 problem 1226

Internal problem ID [11207]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1226
Date solved : Sunday, March 30, 2025 at 07:46:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 23

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

\[ y = c_1 \operatorname {LegendreP}\left (v -1, i x \right )+c_2 \operatorname {LegendreQ}\left (v -1, i x \right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 30

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

\[ y(x)\to c_1 \operatorname {LegendreP}(v-1,i x)+c_2 \operatorname {LegendreQ}(v-1,i x) \]

✗ Sympy

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

False