54.3.175 problem 1189

Internal problem ID [12470]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1189
Date solved : Wednesday, October 01, 2025 at 01:45:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{m}+c \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 80

ode:=x^2*diff(diff(y(x),x),x)+a*x*diff(y(x),x)+(b*x^m+c)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{-\frac {a}{2}} \sqrt {x}\, \left (\operatorname {BesselY}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{m}, \frac {2 \sqrt {b}\, x^{\frac {m}{2}}}{m}\right ) c_2 +\operatorname {BesselJ}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{m}, \frac {2 \sqrt {b}\, x^{\frac {m}{2}}}{m}\right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.085 (sec). Leaf size: 168

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

\begin{align*} y(x)&\to m^{\frac {a-1}{m}} b^{-\frac {a-1}{2 m}} \left (x^m\right )^{-\frac {a-1}{2 m}} \left (c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {a^2-2 a-4 c+1}}{m}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {a^2-2 a-4 c+1}}{m},\frac {2 \sqrt {b} \sqrt {x^m}}{m}\right )+c_2 \operatorname {Gamma}\left (\frac {m+\sqrt {a^2-2 a-4 c+1}}{m}\right ) \operatorname {BesselJ}\left (\frac {\sqrt {a^2-2 a-4 c+1}}{m},\frac {2 \sqrt {b} \sqrt {x^m}}{m}\right )\right ) \end{align*}

✗ Sympy

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

TypeError : invalid input: 1 - a