81.2.3 problem 3-3
Internal
problem
ID
[21492]
Book
:
The
Differential
Equations
Problem
Solver.
VOL.
I.
M.
Fogiel
director.
REA,
NY.
1978.
ISBN
78-63609
Section
:
Chapter
3.
Exact
differential
equations.
Page
42.
Problem
number
:
3-3
Date
solved
:
Thursday, October 02, 2025 at 07:41:56 PM
CAS
classification
:
[[_homogeneous, `class G`], _exact, _rational]
\begin{align*} \left (x +y^{2}\right ) y^{\prime }+y&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 224
ode:=(x+y(x)^2)*diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (12 c_1 +4 \sqrt {4 x^{3}+9 c_1^{2}}\right )^{{2}/{3}}-4 x}{2 \left (12 c_1 +4 \sqrt {4 x^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {i \sqrt {3}\, \left (12 c_1 +4 \sqrt {4 x^{3}+9 c_1^{2}}\right )^{{2}/{3}}+4 i \sqrt {3}\, x +\left (12 c_1 +4 \sqrt {4 x^{3}+9 c_1^{2}}\right )^{{2}/{3}}-4 x}{4 \left (12 c_1 +4 \sqrt {4 x^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {i \sqrt {3}\, \left (12 c_1 +4 \sqrt {4 x^{3}+9 c_1^{2}}\right )^{{2}/{3}}+4 i \sqrt {3}\, x -\left (12 c_1 +4 \sqrt {4 x^{3}+9 c_1^{2}}\right )^{{2}/{3}}+4 x}{4 \left (12 c_1 +4 \sqrt {4 x^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 0.689 (sec). Leaf size: 263
ode=(x+y[x]^2)*D[y[x],x] +y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt [3]{\sqrt {4 x^3+9 c_1{}^2}+3 c_1}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x}{\sqrt [3]{\sqrt {4 x^3+9 c_1{}^2}+3 c_1}}\\ y(x)&\to \frac {i 2^{2/3} \left (\sqrt {3}+i\right ) \left (\sqrt {4 x^3+9 c_1{}^2}+3 c_1\right ){}^{2/3}+\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x}{4 \sqrt [3]{\sqrt {4 x^3+9 c_1{}^2}+3 c_1}}\\ y(x)&\to \frac {\left (1-i \sqrt {3}\right ) x}{2^{2/3} \sqrt [3]{\sqrt {4 x^3+9 c_1{}^2}+3 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {4 x^3+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}}\\ y(x)&\to 0 \end{align*}
✓ Sympy. Time used: 6.895 (sec). Leaf size: 209
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((x + y(x)**2)*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \sqrt [3]{2} \left (\frac {x}{\sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + 4 x^{3}}}} - \frac {\sqrt [3]{2} \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + 4 x^{3}}}}{2}\right ), \ y{\left (x \right )} = \sqrt [3]{2} \left (\frac {2 x}{\left (-1 - \sqrt {3} i\right ) \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + 4 x^{3}}}} - \frac {\sqrt [3]{2} \left (-1 - \sqrt {3} i\right ) \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + 4 x^{3}}}}{4}\right ), \ y{\left (x \right )} = \sqrt [3]{2} \left (\frac {2 x}{\left (-1 + \sqrt {3} i\right ) \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + 4 x^{3}}}} - \frac {\sqrt [3]{2} \left (-1 + \sqrt {3} i\right ) \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + 4 x^{3}}}}{4}\right )\right ]
\]