23.3.231 problem 233

Internal problem ID [5945]
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 : 233
Date solved : Tuesday, September 30, 2025 at 02:06:38 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 31

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

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

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 46

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

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

✓ Sympy. Time used: 0.086 (sec). Leaf size: 48

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

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