60.1.175 problem 178
Internal
problem
ID
[10189]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
178
Date
solved
:
Sunday, March 30, 2025 at 03:27:04 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} 2 x \left (x^{2}-1\right ) y^{\prime }+2 \left (x^{2}-1\right ) y^{2}-\left (3 x^{2}-5\right ) y+x^{2}-3&=0 \end{align*}
✓ Maple. Time used: 0.008 (sec). Leaf size: 103
ode:=2*x*(x^2-1)*diff(y(x),x)+2*(x^2-1)*y(x)^2-(3*x^2-5)*y(x)+x^2-3 = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-2 \operatorname {EllipticF}\left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x}\, \sqrt {1-x}\, \sqrt {x +1}\, \sqrt {2}+\sqrt {x}\, \sqrt {x -1}\, \sqrt {x +1}\, c_1 -2 x}{\sqrt {x +1}\, \left (-2 \operatorname {EllipticF}\left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x}\, \sqrt {2}\, \sqrt {1-x}+c_1 \sqrt {x -1}\, \sqrt {x}\right )}
\]
✓ Mathematica. Time used: 0.372 (sec). Leaf size: 174
ode=2*x*(x^2-1)*D[y[x],x] + 2*(x^2-1)*y[x]^2 - (3*x^2-5)*y[x] + x^2 - 3==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to 1+\frac {\exp \left (\int _1^x\frac {K[1]^2+1}{2 K[1]-2 K[1]^3}dK[1]\right )}{-\int _1^x-\frac {\exp \left (\int _1^{K[2]}\frac {K[1]^2+1}{2 K[1]-2 K[1]^3}dK[1]\right )}{K[2]}dK[2]+c_1} \\
y(x)\to 1 \\
y(x)\to 1-\frac {\exp \left (\int _1^x\frac {K[1]^2+1}{2 K[1]-2 K[1]^3}dK[1]\right )}{\int _1^x-\frac {\exp \left (\int _1^{K[2]}\frac {K[1]^2+1}{2 K[1]-2 K[1]^3}dK[1]\right )}{K[2]}dK[2]} \\
\end{align*}
✓ Sympy. Time used: 36.258 (sec). Leaf size: 218
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2 + 2*x*(x**2 - 1)*Derivative(y(x), x) + (2*x**2 - 2)*y(x)**2 - (3*x**2 - 5)*y(x) - 3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \frac {- 16 x + 6 \sqrt {x \left (x^{2} - 1\right )} \int \frac {1}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right )^{2} \left (x + 1\right )^{2}}\, dx + 3 \sqrt {x \left (x^{2} - 1\right )} \int \frac {1}{x^{2} \sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right )^{2} \left (x + 1\right )^{2}}\, dx + 3 \sqrt {x \left (x^{2} - 1\right )} \int \frac {x^{2}}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right )^{2} \left (x + 1\right )^{2}}\, dx}{3 \sqrt {x \left (x^{2} - 1\right )} \left (2 \int \frac {1}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right )^{2} \left (x + 1\right )^{2}}\, dx + \int \frac {1}{x^{2} \sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right )^{2} \left (x + 1\right )^{2}}\, dx + \int \frac {x^{2}}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right )^{2} \left (x + 1\right )^{2}}\, dx\right )}
\]