54.3.393 problem 1410

Internal problem ID [12688]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1410
Date solved : Friday, October 03, 2025 at 03:46:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.094 (sec). Leaf size: 250

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

\[ y = x^{\frac {1}{2}+\frac {q}{2}} \left (c_1 \,x^{\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {p +q +\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}, \frac {p +q +\sqrt {q^{2}+2 q +4 s +1}-\sqrt {p^{2}-2 p -4 r +1}}{2 b}\right ], \left [1+\frac {\sqrt {q^{2}+2 q +4 s +1}}{b}\right ], a \,x^{b}\right )+c_2 \,x^{-\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}} \operatorname {hypergeom}\left (\left [-\frac {-p -q +\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}, \frac {p +q -\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}\right ], \left [1-\frac {\sqrt {q^{2}+2 q +4 s +1}}{b}\right ], a \,x^{b}\right )\right ) \]

✓ Mathematica. Time used: 0.131 (sec). Leaf size: 405

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

\begin{align*} y(x)&\to c_1 i^{\frac {-\sqrt {q^2+2 q+4 s+1}+q+1}{b}} a^{\frac {-\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \left (x^b\right )^{\frac {-\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \operatorname {Hypergeometric2F1}\left (\frac {p+q-\sqrt {p^2-2 p-4 r+1}-\sqrt {q^2+2 q+4 s+1}}{2 b},\frac {p+q+\sqrt {p^2-2 p-4 r+1}-\sqrt {q^2+2 q+4 s+1}}{2 b},1-\frac {\sqrt {q^2+2 q+4 s+1}}{b},a x^b\right )+c_2 i^{\frac {\sqrt {q^2+2 q+4 s+1}+q+1}{b}} a^{\frac {\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \left (x^b\right )^{\frac {\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \operatorname {Hypergeometric2F1}\left (\frac {p+q-\sqrt {p^2-2 p-4 r+1}+\sqrt {q^2+2 q+4 s+1}}{2 b},\frac {p+q+\sqrt {p^2-2 p-4 r+1}+\sqrt {q^2+2 q+4 s+1}}{2 b},\frac {b+\sqrt {q^2+2 q+4 s+1}}{b},a x^b\right ) \end{align*}

✗ Sympy

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

TypeError : Symbol object cannot be interpreted as an integer