23.1.378 problem 363
Internal
problem
ID
[4985]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
1.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
FIRST
DEGREE,
page
223
Problem
number
:
363
Date
solved
:
Tuesday, September 30, 2025 at 09:08:50 AM
CAS
classification
:
[_linear]
\begin{align*} x \left (-x^{2}+1\right ) y^{\prime }&=x^{3} \left (-x^{2}+1\right )+\left (-2 x^{2}+1\right ) y \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 24
ode:=x*(-x^2+1)*diff(y(x),x) = x^3*(-x^2+1)+(-2*x^2+1)*y(x);
dsolve(ode,y(x), singsol=all);
\[
y = \sqrt {x +1}\, \sqrt {x -1}\, c_1 x +x^{3}-x
\]
✓ Mathematica. Time used: 0.066 (sec). Leaf size: 82
ode=x*(1-x^2)*D[y[x],x]==x^3*(1-x^2)+(1-2*x^2)*y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \exp \left (\int _1^x\frac {1-2 K[1]^2}{K[1]-K[1]^3}dK[1]\right ) \left (\int _1^x\exp \left (-\int _1^{K[2]}\frac {1-2 K[1]^2}{K[1]-K[1]^3}dK[1]\right ) K[2]^2dK[2]+c_1\right ) \end{align*}
✓ Sympy. Time used: 19.690 (sec). Leaf size: 328
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**3*(1 - x**2) + x*(1 - x**2)*Derivative(y(x), x) - (1 - 2*x**2)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \begin {cases} \tilde {\infty } x^{3} + \tilde {\infty } x \sqrt {x^{2} - 1} \int \frac {y{\left (x \right )}}{\left (\left (x - 1\right ) \left (x + 1\right )\right )^{\frac {3}{2}}}\, dx + \tilde {\infty } x \sqrt {x^{2} - 1} \int \frac {y{\left (x \right )}}{x^{2} \left (\left (x - 1\right ) \left (x + 1\right )\right )^{\frac {3}{2}}}\, dx + \tilde {\infty } x \sqrt {x^{2} - 1} + \tilde {\infty } x & \text {for}\: x > -1 \wedge \left |{x^{2}}\right | > 1 \wedge x < 1 \\\frac {C_{1} \sqrt {1 - x^{2}} \sqrt {x^{2} - 1}}{2 x \sqrt {1 - x^{2}} + 2 i x \sqrt {x^{2} - 1}} + \frac {x^{2} \sqrt {1 - x^{2}}}{2 x \sqrt {1 - x^{2}} + 2 i x \sqrt {x^{2} - 1}} + \frac {2 \sqrt {1 - x^{2}} \sqrt {x^{2} - 1} \int \frac {y{\left (x \right )}}{\left (\left (x - 1\right ) \left (x + 1\right )\right )^{\frac {3}{2}}}\, dx}{2 x \sqrt {1 - x^{2}} + 2 i x \sqrt {x^{2} - 1}} - \frac {\sqrt {1 - x^{2}} \sqrt {x^{2} - 1} \int \frac {y{\left (x \right )}}{x^{2} \left (\left (x - 1\right ) \left (x + 1\right )\right )^{\frac {3}{2}}}\, dx}{2 x \sqrt {1 - x^{2}} + 2 i x \sqrt {x^{2} - 1}} - \frac {\sqrt {1 - x^{2}}}{2 x \sqrt {1 - x^{2}} + 2 i x \sqrt {x^{2} - 1}} & \text {for}\: x > -1 \wedge x < 1 \end {cases}
\]