32.4.26 problem Recognizable Exact Differential equations. Integrating factors. Exercise 10.18, page 90
Internal
problem
ID
[5837]
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
10.18,
page
90
Date
solved
:
Sunday, March 30, 2025 at 10:19:11 AM
CAS
classification
:
[_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
\begin{align*} 2 x y+\left (x^{2}+y^{2}+a \right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 309
ode:=2*x*y(x)+(x^2+y(x)^2+a)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-12 c_1 +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 a^{2} x^{2}+4 a^{3}+9 c_1^{2}}\right )^{{2}/{3}}-4 x^{2}-4 a}{2 \left (-12 c_1 +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 a^{2} x^{2}+4 a^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {\left (\frac {i \sqrt {3}}{4}+\frac {1}{4}\right ) \left (-12 c_1 +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 a^{2} x^{2}+4 a^{3}+9 c_1^{2}}\right )^{{2}/{3}}+\left (i \sqrt {3}-1\right ) \left (x^{2}+a \right )}{\left (-12 c_1 +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 a^{2} x^{2}+4 a^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-12 c_1 +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 a^{2} x^{2}+4 a^{3}+9 c_1^{2}}\right )^{{2}/{3}}}{4}+\left (x^{2}+a \right ) \left (1+i \sqrt {3}\right )}{\left (-12 c_1 +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 a^{2} x^{2}+4 a^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 4.811 (sec). Leaf size: 299
ode=(2*x*y[x])+(x^2+y[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+9 c_1{}^2}+3 c_1\right ){}^{2/3}-2 a-2 x^2}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+9 c_1{}^2}+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+9 c_1{}^2}+3 c_1}}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+9 c_1{}^2}+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+9 c_1{}^2}+3 c_1}}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}} \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 26.062 (sec). Leaf size: 221
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(2*x*y(x) + (a + x**2 + y(x)**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} - \left (- a - x^{2}\right )^{3}}}}{2} - \frac {2 \left (- a - x^{2}\right )}{\left (-1 - \sqrt {3} i\right ) \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} - \left (- a - x^{2}\right )^{3}}}}, \ y{\left (x \right )} = - \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} - \left (- a - x^{2}\right )^{3}}}}{2} - \frac {2 \left (- a - x^{2}\right )}{\left (-1 + \sqrt {3} i\right ) \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} - \left (- a - x^{2}\right )^{3}}}}, \ y{\left (x \right )} = - \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} - \left (- a - x^{2}\right )^{3}}} - \frac {- a - x^{2}}{\sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} - \left (- a - x^{2}\right )^{3}}}}\right ]
\]