77.1.142 problem thm 19 (page 244)

Internal problem ID [17961]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : thm 19 (page 244)
Date solved : Monday, March 31, 2025 at 04:52:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\frac {y^{\prime }}{x}+\left (1-\frac {m^{2}}{x^{2}}\right ) y&=0 \end{align*}

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

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

\[ y = x \left (c_1 \operatorname {BesselJ}\left (\sqrt {m^{2}+1}, x\right )+c_2 \operatorname {BesselY}\left (\sqrt {m^{2}+1}, x\right )\right ) \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 36

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

\[ y(x)\to x \left (c_1 \operatorname {BesselJ}\left (\sqrt {m^2+1},x\right )+c_2 \operatorname {BesselY}\left (\sqrt {m^2+1},x\right )\right ) \]

✓ Sympy. Time used: 0.209 (sec). Leaf size: 27

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

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