77.1.106 problem 134 (page 195)

Internal problem ID [17917]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 134 (page 195)
Date solved : Thursday, March 13, 2025 at 11:10:32 AM
CAS classification : [_Lienard]

\begin{align*} y^{\prime \prime }+\frac {2 y^{\prime }}{x}+y&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 17

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

\[ y = \frac {\sin \left (x \right ) c_{1} +\cos \left (x \right ) c_{2}}{x} \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 37

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

\[ y(x)\to \frac {2 c_1 e^{-i x}-i c_2 e^{i x}}{2 x} \]

✓ Sympy. Time used: 0.183 (sec). Leaf size: 20

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

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