23.3.491 problem 497
Internal
problem
ID
[6205]
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
:
497
Date
solved
:
Tuesday, September 30, 2025 at 02:36:18 PM
CAS
classification
:
[[_2nd_order, _missing_y]]
\begin{align*} -y^{\prime }+x \left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 15
ode:=-diff(y(x),x)+x*(-x^2+1)*diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = c_1 +\sqrt {x^{2}-1}\, c_2
\]
✓ Mathematica. Time used: 0.011 (sec). Leaf size: 23
ode=-D[y[x],x] + x*(1 - x^2)*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to c_2-c_1 \sqrt {1-x^2} \end{align*}
✓ Sympy. Time used: 0.283 (sec). Leaf size: 144
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(1 - x**2)*Derivative(y(x), (x, 2)) - Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} + x^{- \left (- \left (\operatorname {re}{\left (x\right )}\right )^{2} + \left (\operatorname {im}{\left (x\right )}\right )^{2} + 2\right ) \operatorname {re}{\left (\frac {1}{\left (x - 1\right ) \left (x + 1\right )}\right )} - 2 \operatorname {re}{\left (x\right )} \operatorname {im}{\left (x\right )} \operatorname {im}{\left (\frac {1}{\left (x - 1\right ) \left (x + 1\right )}\right )}} \left (C_{2} \sin {\left (\log {\left (x \right )} \left |{\left (- \left (\operatorname {re}{\left (x\right )}\right )^{2} + \left (\operatorname {im}{\left (x\right )}\right )^{2} + 2\right ) \operatorname {im}{\left (\frac {1}{\left (x - 1\right ) \left (x + 1\right )}\right )} - 2 \operatorname {re}{\left (x\right )} \operatorname {re}{\left (\frac {1}{\left (x - 1\right ) \left (x + 1\right )}\right )} \operatorname {im}{\left (x\right )}}\right | \right )} + C_{3} \cos {\left (\left (\left (- \left (\operatorname {re}{\left (x\right )}\right )^{2} + \left (\operatorname {im}{\left (x\right )}\right )^{2} + 2\right ) \operatorname {im}{\left (\frac {1}{\left (x - 1\right ) \left (x + 1\right )}\right )} - 2 \operatorname {re}{\left (x\right )} \operatorname {re}{\left (\frac {1}{\left (x - 1\right ) \left (x + 1\right )}\right )} \operatorname {im}{\left (x\right )}\right ) \log {\left (x \right )} \right )}\right )
\]