23.3.473 problem 479

Internal problem ID [6187]
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 : 479
Date solved : Tuesday, September 30, 2025 at 02:35:58 PM
CAS classification : [[_2nd_order, _missing_y]]

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

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

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

\[ y = c_1 +\left (\ln \left (a x -1\right )-\ln \left (a x +1\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 19

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

\begin{align*} y(x)&\to c_2-\frac {c_1 \text {arctanh}(a x)}{a} \end{align*}

✓ Sympy. Time used: 0.202 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = C_{1} - \frac {C_{2} \log {\left (x - \frac {1}{a} \right )}}{2 a} + \frac {C_{2} \log {\left (x + \frac {1}{a} \right )}}{2 a} \]