32.6.36 problem Exercise 12.36, page 103
Internal
problem
ID
[5901]
Book
:
Ordinary
Differential
Equations,
By
Tenenbaum
and
Pollard.
Dover,
NY
1963
Section
:
Chapter
2.
Special
types
of
differential
equations
of
the
first
kind.
Lesson
12,
Miscellaneous
Methods
Problem
number
:
Exercise
12.36,
page
103
Date
solved
:
Sunday, March 30, 2025 at 10:24:38 AM
CAS
classification
:
[_exact, _rational]
\begin{align*} \left (x^{2}+y^{2}+1\right ) y^{\prime }+2 x y+x^{2}+3&=0 \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 366
ode:=(1+x^2+y(x)^2)*diff(y(x),x)+2*x*y(x)+x^2+3 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-4 x^{3}-12 c_1 -36 x +4 \sqrt {5 x^{6}+6 c_1 \,x^{3}+30 x^{4}+9 c_1^{2}+54 c_1 x +93 x^{2}+4}\right )^{{2}/{3}}-4 x^{2}-4}{2 \left (-4 x^{3}-12 c_1 -36 x +4 \sqrt {5 x^{6}+6 c_1 \,x^{3}+30 x^{4}+9 c_1^{2}+54 c_1 x +93 x^{2}+4}\right )^{{1}/{3}}} \\
y &= -\frac {\left (\frac {i \sqrt {3}}{4}+\frac {1}{4}\right ) \left (-4 x^{3}-12 c_1 -36 x +4 \sqrt {5 x^{6}+6 c_1 \,x^{3}+30 x^{4}+9 c_1^{2}+54 c_1 x +93 x^{2}+4}\right )^{{2}/{3}}+\left (i \sqrt {3}-1\right ) \left (x^{2}+1\right )}{\left (-4 x^{3}-12 c_1 -36 x +4 \sqrt {5 x^{6}+6 c_1 \,x^{3}+30 x^{4}+9 c_1^{2}+54 c_1 x +93 x^{2}+4}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (-4 x^{3}-12 c_1 -36 x +4 \sqrt {5 x^{6}+6 c_1 \,x^{3}+30 x^{4}+9 c_1^{2}+54 c_1 x +93 x^{2}+4}\right )^{{1}/{3}}}{4}+\frac {\left (1+i \sqrt {3}\right ) \left (x^{2}+1\right )}{\left (-4 x^{3}-12 c_1 -36 x +4 \sqrt {5 x^{6}+6 c_1 \,x^{3}+30 x^{4}+9 c_1^{2}+54 c_1 x +93 x^{2}+4}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 5.625 (sec). Leaf size: 411
ode=(x^2+y[x]^2+1)*D[y[x],x]+2*x*y[x]+x^2+3==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{-27 x^3+\sqrt {4 \left (9 x^2+9\right )^3+729 \left (x^3+9 x-3 c_1\right ){}^2}-243 x+81 c_1}}{3 \sqrt [3]{2}}-\frac {3 \sqrt [3]{2} \left (x^2+1\right )}{\sqrt [3]{-27 x^3+\sqrt {4 \left (9 x^2+9\right )^3+729 \left (x^3+9 x-3 c_1\right ){}^2}-243 x+81 c_1}} \\
y(x)\to \frac {3 \left (1+i \sqrt {3}\right ) \left (x^2+1\right )}{2^{2/3} \sqrt [3]{-27 x^3+\sqrt {4 \left (9 x^2+9\right )^3+729 \left (x^3+9 x-3 c_1\right ){}^2}-243 x+81 c_1}}+\frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{-27 x^3+\sqrt {4 \left (9 x^2+9\right )^3+729 \left (x^3+9 x-3 c_1\right ){}^2}-243 x+81 c_1}}{6 \sqrt [3]{2}} \\
y(x)\to \frac {3 \left (1-i \sqrt {3}\right ) \left (x^2+1\right )}{2^{2/3} \sqrt [3]{-27 x^3+\sqrt {4 \left (9 x^2+9\right )^3+729 \left (x^3+9 x-3 c_1\right ){}^2}-243 x+81 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-27 x^3+\sqrt {4 \left (9 x^2+9\right )^3+729 \left (x^3+9 x-3 c_1\right ){}^2}-243 x+81 c_1}}{6 \sqrt [3]{2}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2 + 2*x*y(x) + (x**2 + y(x)**2 + 1)*Derivative(y(x), x) + 3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out