54.3.141 problem 1155

Internal problem ID [12436]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1155
Date solved : Wednesday, October 01, 2025 at 01:45:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (a \,x^{k}-b \left (b -1\right )\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 71

ode:=x^2*diff(diff(y(x),x),x)+(a*x^k-b*(b-1))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.053 (sec). Leaf size: 116

ode=((1 - b)*b + a*x^k)*y[x] + x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.086 (sec). Leaf size: 68

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

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} J_{\frac {2 \sqrt {b \left (b - 1\right ) + \frac {1}{4}}}{k}}\left (\frac {2 \sqrt {a} x^{\frac {k}{2}}}{k}\right ) + C_{2} Y_{\frac {2 \sqrt {b \left (b - 1\right ) + \frac {1}{4}}}{k}}\left (\frac {2 \sqrt {a} x^{\frac {k}{2}}}{k}\right )\right ) \]