23.3.274 problem 276

Internal problem ID [5988]
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 : 276
Date solved : Tuesday, September 30, 2025 at 02:07:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} -\left (i x^{2}+p^{2}\right ) y+x y^{\prime }+x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 29

ode:=-(p^2+I*x^2)*y(x)+x*diff(y(x),x)+x^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {BesselI}\left (p , \left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right )+c_2 \operatorname {BesselK}\left (p , \left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right ) \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 32

ode=-((p^2 + I*x^2)*y[x]) + 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 c_1 \operatorname {BesselJ}\left (p,-(-1)^{3/4} x\right )+c_2 \operatorname {BesselY}\left (p,-(-1)^{3/4} x\right ) \end{align*}

✓ Sympy. Time used: 0.151 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = C_{1} J_{\sqrt {p^{2}}}\left (x \sqrt {- \operatorname {complex}{\left (0,1 \right )}}\right ) + C_{2} Y_{\sqrt {p^{2}}}\left (x \sqrt {- \operatorname {complex}{\left (0,1 \right )}}\right ) \]