32.4.27 problem Recognizable Exact Differential equations. Integrating factors. Exercise 10.19, page 90
Internal
problem
ID
[5838]
Book
:
Ordinary
Differential
Equations,
By
Tenenbaum
and
Pollard.
Dover,
NY
1963
Section
:
Chapter
2.
Special
types
of
differential
equations
of
the
first
kind.
Lesson
10
Problem
number
:
Recognizable
Exact
Differential
equations.
Integrating
factors.
Exercise
10.19,
page
90
Date
solved
:
Sunday, March 30, 2025 at 10:19:12 AM
CAS
classification
:
[_exact, _rational]
\begin{align*} 2 x y+x^{2}+b +\left (y^{2}+x^{2}+a \right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 501
ode:=2*x*y(x)+x^2+b+(x^2+y(x)^2+a)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {-4 x^{2}-4 a +\left (-4 x^{3}-12 b x -12 c_1 +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_1 \,x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 b x c_1 +4 a^{3}+9 c_1^{2}}\right )^{{2}/{3}}}{2 \left (-4 x^{3}-12 b x -12 c_1 +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_1 \,x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 b x c_1 +4 a^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {\left (\frac {i \sqrt {3}}{4}+\frac {1}{4}\right ) \left (-4 x^{3}-12 b x -12 c_1 +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_1 \,x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 b x c_1 +4 a^{3}+9 c_1^{2}}\right )^{{2}/{3}}+\left (i \sqrt {3}-1\right ) \left (x^{2}+a \right )}{\left (-4 x^{3}-12 b x -12 c_1 +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_1 \,x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 b x c_1 +4 a^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-4 x^{3}-12 b x -12 c_1 +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_1 \,x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 b x c_1 +4 a^{3}+9 c_1^{2}}\right )^{{2}/{3}}}{4}+\left (x^{2}+a \right ) \left (1+i \sqrt {3}\right )}{\left (-4 x^{3}-12 b x -12 c_1 +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_1 \,x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 b x c_1 +4 a^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 7.538 (sec). Leaf size: 396
ode=(2*x*y[x]+x^2+b)+(y[x]^2+x^2+a)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{2} \left (\sqrt {4 \left (a+x^2\right )^3+\left (3 b x+x^3-3 c_1\right ){}^2}-3 b x-x^3+3 c_1\right ){}^{2/3}-2 a-2 x^2}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+\left (3 b x+x^3-3 c_1\right ){}^2}-3 b x-x^3+3 c_1}} \\
y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (a+x^2\right )}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+\left (3 b x+x^3-3 c_1\right ){}^2}-3 b x-x^3+3 c_1}}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+\left (3 b x+x^3-3 c_1\right ){}^2}-3 b x-x^3+3 c_1}}{2 \sqrt [3]{2}} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (a+x^2\right )}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+\left (3 b x+x^3-3 c_1\right ){}^2}-3 b x-x^3+3 c_1}}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+\left (3 b x+x^3-3 c_1\right ){}^2}-3 b x-x^3+3 c_1}}{2 \sqrt [3]{2}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(b + x**2 + 2*x*y(x) + (a + x**2 + y(x)**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out