23.3.450 problem 455

Internal problem ID [6164]
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 : 455
Date solved : Tuesday, September 30, 2025 at 02:23:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} -\left (a^{2} x^{2}+1\right ) y+4 x y^{\prime }+4 x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 35

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

\begin{align*} y(x)&\to \frac {e^{-\frac {a x}{2}} \left (c_2 e^{a x}+a c_1\right )}{a \sqrt {x}} \end{align*}

✓ Sympy. Time used: 0.145 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + 4*x*Derivative(y(x), x) + (-a**2*x**2 - 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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