54.3.282 problem 1299

Internal problem ID [12577]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1299
Date solved : Wednesday, October 01, 2025 at 02:09:53 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 23

ode:=(a^2*x^2-1)*diff(diff(y(x),x),x)+2*a^2*x*diff(y(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: 0.013 (sec). Leaf size: 30

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 \int _1^x\frac {c_1}{a^2 K[1]^2-1}dK[1]+c_2 \end{align*}

✓ Sympy. Time used: 0.212 (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)
 

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