54.3.322 problem 1339

Internal problem ID [12617]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1339
Date solved : Friday, October 03, 2025 at 03:43:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.049 (sec). Leaf size: 76

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

\[ y = \left (x^{d} \operatorname {hypergeom}\left (\left [c , 1-b +c \right ], \left [1+c +d \right ], -a x \right ) c_1 +x^{-c} \operatorname {hypergeom}\left (\left [-d , 1-b -d \right ], \left [1-c -d \right ], -a x \right ) c_2 \right ) \left (a x +1\right )^{-b +c -d} \]

✓ Mathematica. Time used: 0.178 (sec). Leaf size: 66

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

\begin{align*} y(x)&\to c_1 a^{-c} x^{-c} \operatorname {Hypergeometric2F1}(1-c,b-c,-c-d+1,-a x)+c_2 a^d x^d \operatorname {Hypergeometric2F1}(d+1,b+d,c+d+1,-a x) \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None