15.8.6 problem 6
Internal
problem
ID
[3009]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
12,
page
46
Problem
number
:
6
Date
solved
:
Sunday, March 30, 2025 at 01:05:43 AM
CAS
classification
:
[_exact, _rational]
\begin{align*} 2 x y-2 x y^{3}+x^{3}+\left (x^{2}+y^{2}-3 x^{2} y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 536
ode:=2*x*y(x)-2*x*y(x)^3+x^3+(x^2+y(x)^2-3*x^2*y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {12 x^{4}-4 x^{2}+{\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_1 -64\right ) x^{6}-72 c_1 \,x^{4}+432 c_1^{2} x^{2}-144 c_1^{2}}{3 x^{2}-1}}+12 c_1 \right ) \left (3 x^{2}-1\right )^{2}\right )}^{{2}/{3}}}{{\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_1 -64\right ) x^{6}-72 c_1 \,x^{4}+432 c_1^{2} x^{2}-144 c_1^{2}}{3 x^{2}-1}}+12 c_1 \right ) \left (3 x^{2}-1\right )^{2}\right )}^{{1}/{3}} \left (6 x^{2}-2\right )} \\
y &= \frac {\left (-i \sqrt {3}-1\right ) {\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_1 -64\right ) x^{6}-72 c_1 \,x^{4}+432 c_1^{2} x^{2}-144 c_1^{2}}{3 x^{2}-1}}+12 c_1 \right ) \left (3 x^{2}-1\right )^{2}\right )}^{{1}/{3}}+\frac {12 \left (i \sqrt {3}-1\right ) \left (x^{2}-\frac {1}{3}\right ) x^{2}}{{\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_1 -64\right ) x^{6}-72 c_1 \,x^{4}+432 c_1^{2} x^{2}-144 c_1^{2}}{3 x^{2}-1}}+12 c_1 \right ) \left (3 x^{2}-1\right )^{2}\right )}^{{1}/{3}}}}{12 x^{2}-4} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) {\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_1 -64\right ) x^{6}-72 c_1 \,x^{4}+432 c_1^{2} x^{2}-144 c_1^{2}}{3 x^{2}-1}}+12 c_1 \right ) \left (3 x^{2}-1\right )^{2}\right )}^{{2}/{3}}}{4}+3 \left (-i \sqrt {3}-1\right ) \left (x^{2}-\frac {1}{3}\right ) x^{2}}{{\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_1 -64\right ) x^{6}-72 c_1 \,x^{4}+432 c_1^{2} x^{2}-144 c_1^{2}}{3 x^{2}-1}}+12 c_1 \right ) \left (3 x^{2}-1\right )^{2}\right )}^{{1}/{3}} \left (3 x^{2}-1\right )} \\
\end{align*}
✓ Mathematica. Time used: 60.19 (sec). Leaf size: 723
ode=(2*x*y[x]-2*x*y[x]^3+x^3)+(x^2+y[x]^2-3*x^2*y[x]^2)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {12 x^4-4 x^2+\left (27 x^8-18 x^6+3 (1+36 c_1) x^4-72 c_1 x^2+\sqrt {\left (3 x^2-1\right )^3 \left (27 x^{10}-9 x^8+8 (-8+27 c_1) x^6-72 c_1 x^4+432 c_1{}^2 x^2-144 c_1{}^2\right )}+12 c_1\right ){}^{2/3}}{2 \left (3 x^2-1\right ) \sqrt [3]{27 x^8-18 x^6+3 (1+36 c_1) x^4-72 c_1 x^2+\sqrt {\left (3 x^2-1\right )^3 \left (27 x^{10}-9 x^8+8 (-8+27 c_1) x^6-72 c_1 x^4+432 c_1{}^2 x^2-144 c_1{}^2\right )}+12 c_1}} \\
y(x)\to \frac {-12 i \left (\sqrt {3}-i\right ) x^4+\left (4+4 i \sqrt {3}\right ) x^2+i \left (\sqrt {3}+i\right ) \left (27 x^8-18 x^6+3 (1+36 c_1) x^4-72 c_1 x^2+\sqrt {\left (3 x^2-1\right )^3 \left (27 x^{10}-9 x^8+8 (-8+27 c_1) x^6-72 c_1 x^4+432 c_1{}^2 x^2-144 c_1{}^2\right )}+12 c_1\right ){}^{2/3}}{4 \left (3 x^2-1\right ) \sqrt [3]{27 x^8-18 x^6+3 (1+36 c_1) x^4-72 c_1 x^2+\sqrt {\left (3 x^2-1\right )^3 \left (27 x^{10}-9 x^8+8 (-8+27 c_1) x^6-72 c_1 x^4+432 c_1{}^2 x^2-144 c_1{}^2\right )}+12 c_1}} \\
y(x)\to \frac {12 i \left (\sqrt {3}+i\right ) x^4+\left (4-4 i \sqrt {3}\right ) x^2-i \left (\sqrt {3}-i\right ) \left (27 x^8-18 x^6+3 (1+36 c_1) x^4-72 c_1 x^2+\sqrt {\left (3 x^2-1\right )^3 \left (27 x^{10}-9 x^8+8 (-8+27 c_1) x^6-72 c_1 x^4+432 c_1{}^2 x^2-144 c_1{}^2\right )}+12 c_1\right ){}^{2/3}}{4 \left (3 x^2-1\right ) \sqrt [3]{27 x^8-18 x^6+3 (1+36 c_1) x^4-72 c_1 x^2+\sqrt {\left (3 x^2-1\right )^3 \left (27 x^{10}-9 x^8+8 (-8+27 c_1) x^6-72 c_1 x^4+432 c_1{}^2 x^2-144 c_1{}^2\right )}+12 c_1}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**3 - 2*x*y(x)**3 + 2*x*y(x) + (-3*x**2*y(x)**2 + x**2 + y(x)**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out