54.3.351 problem 1368

Internal problem ID [12646]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1368
Date solved : Friday, October 03, 2025 at 03:45:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 69

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

\[ y = \left (x^{2}+1\right )^{\frac {1}{2}-\frac {a}{4}} \left (\operatorname {LegendreQ}\left (\frac {a}{2}-1, \frac {\sqrt {a^{2}-4 a +4 b +4}}{2}, i x \right ) c_2 +\operatorname {LegendreP}\left (\frac {a}{2}-1, \frac {\sqrt {a^{2}-4 a +4 b +4}}{2}, i x \right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 92

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

\begin{align*} y(x)&\to \left (x^2+1\right )^{\frac {1}{2}-\frac {a}{4}} \left (c_1 P_{\frac {a-2}{2}}^{\frac {1}{2} \sqrt {a^2-4 a+4 b+4}}(i x)+c_2 Q_{\frac {a-2}{2}}^{\frac {1}{2} \sqrt {a^2-4 a+4 b+4}}(i x)\right ) \end{align*}

✗ Sympy

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

False