23.3.257 problem 259

Internal problem ID [5971]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 259
Date solved : Tuesday, September 30, 2025 at 02:07:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 71

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

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

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 116

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

✓ Sympy. Time used: 0.081 (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*(a - 1) + b*x**k)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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