55.29.1 problem 61

Internal problem ID [13834]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 61
Date solved : Thursday, October 02, 2025 at 08:07:31 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 27

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

\[ y = c_1 \sin \left (2 \sqrt {x}\, \sqrt {a}\right )+c_2 \cos \left (2 \sqrt {x}\, \sqrt {a}\right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 38

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

\begin{align*} y(x)&\to c_1 \cos \left (2 \sqrt {a} \sqrt {x}\right )+c_2 \sin \left (2 \sqrt {a} \sqrt {x}\right ) \end{align*}

✓ Sympy. Time used: 0.105 (sec). Leaf size: 41

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

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