23.3.185 problem 187

Internal problem ID [5899]
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 : 187
Date solved : Friday, October 03, 2025 at 01:45:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left ({\mathrm e}^{x^{2}}-k^{2}\right ) x^{3} y-y^{\prime }+x y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 25

ode:=(exp(x^2)-k^2)*x^3*y(x)-diff(y(x),x)+x*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {BesselJ}\left (k , {\mathrm e}^{\frac {x^{2}}{2}}\right )+c_2 \operatorname {BesselY}\left (k , {\mathrm e}^{\frac {x^{2}}{2}}\right ) \]

✓ Mathematica. Time used: 0.409 (sec). Leaf size: 46

ode=(E^(x^2) - k^2)*x^3*y[x] - D[y[x],x] + x*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
k = symbols("k") 
y = Function("y") 
ode = Eq(x**3*(-k**2 + exp(x**2))*y(x) + x*Derivative(y(x), (x, 2)) - Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False