75.24.9 problem 749

Internal problem ID [17142]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 18.2. Expanding a solution in generalized power series. Bessels equation. Exercises page 177
Problem number : 749
Date solved : Thursday, March 13, 2025 at 09:17:35 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x y^{\prime \prime }+\frac {y^{\prime }}{2}+\frac {y}{4}&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 17

ode:=x*diff(diff(y(x),x),x)+1/2*diff(y(x),x)+1/4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 24

ode=x*D[y[x],{x,2}]+1/2*D[y[x],x]+1/4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \cos \left (\sqrt {x}\right )+c_2 \sin \left (\sqrt {x}\right ) \]

✓ Sympy. Time used: 0.168 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + y(x)/4 + Derivative(y(x), x)/2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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