23.3.184 problem 186

Internal problem ID [5898]
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 : 186
Date solved : Tuesday, September 30, 2025 at 02:05:55 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} -a^{2} x^{3} y-y^{\prime }+x y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

ode:=-a^2*x^3*y(x)-diff(y(x),x)+x*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \sinh \left (\frac {a \,x^{2}}{2}\right )+c_2 \cosh \left (\frac {a \,x^{2}}{2}\right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 33

ode=-(a^2*x^3*y[x]) - D[y[x],x] + x*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \cosh \left (\frac {a x^2}{2}\right )+i c_2 \sinh \left (\frac {a x^2}{2}\right ) \end{align*}

✓ Sympy. Time used: 0.115 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a**2*x**3*y(x) + x*Derivative(y(x), (x, 2)) - Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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