60.3.153 problem 1167

Internal problem ID [11149]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1167
Date solved : Wednesday, March 05, 2025 at 01:43:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.090 (sec). Leaf size: 63

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

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

✓ Mathematica. Time used: 0.131 (sec). Leaf size: 130

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

\[ y(x)\to m^{-2/m} a^{\frac {1}{m}} \left (x^m\right )^{\frac {1}{m}} \left (c_1 \operatorname {Gamma}\left (1-\frac {2 i \sqrt {b-1}}{m}\right ) \operatorname {BesselJ}\left (-\frac {2 i \sqrt {b-1}}{m},\frac {2 \sqrt {a} \sqrt {x^m}}{m}\right )+c_2 \operatorname {Gamma}\left (\frac {2 i \sqrt {b-1}}{m}+1\right ) \operatorname {BesselJ}\left (\frac {2 i \sqrt {b-1}}{m},\frac {2 \sqrt {a} \sqrt {x^m}}{m}\right )\right ) \]

✓ Sympy. Time used: 0.257 (sec). Leaf size: 54

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

\[ y{\left (x \right )} = x \left (C_{1} J_{\frac {2 \sqrt {1 - b}}{m}}\left (\frac {2 \sqrt {a} x^{\frac {m}{2}}}{m}\right ) + C_{2} Y_{\frac {2 \sqrt {1 - b}}{m}}\left (\frac {2 \sqrt {a} x^{\frac {m}{2}}}{m}\right )\right ) \]