76.4.19 problem 25
Internal
problem
ID
[17342]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
2.
First
order
differential
equations.
Section
2.6
(Exact
equations
and
integrating
factors).
Problems
at
page
100
Problem
number
:
25
Date
solved
:
Monday, March 31, 2025 at 03:55:21 PM
CAS
classification
:
[[_homogeneous, `class D`], _rational]
\begin{align*} 3 x^{2} y+2 x y+y^{3}+\left (x^{2}+y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.019 (sec). Leaf size: 293
ode:=3*x^2*y(x)+2*x*y(x)+y(x)^3+(x^2+y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {{\mathrm e}^{-3 x} 2^{{1}/{3}} \left (x^{2} {\mathrm e}^{6 x} c_1^{2}-\frac {2^{{1}/{3}} {\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{2}/{3}}}{2}\right )}{{\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{1}/{3}} c_1} \\
y &= -\frac {\left (\frac {{\mathrm e}^{-3 x} 2^{{1}/{3}} {\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{2}/{3}} \left (1+i \sqrt {3}\right )}{2}+{\mathrm e}^{3 x} x^{2} c_1^{2} \left (i \sqrt {3}-1\right )\right ) 2^{{1}/{3}}}{2 {\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{1}/{3}} c_1} \\
y &= \frac {\left (\frac {{\mathrm e}^{-3 x} 2^{{1}/{3}} {\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{2}/{3}} \left (i \sqrt {3}-1\right )}{2}+{\mathrm e}^{3 x} x^{2} c_1^{2} \left (1+i \sqrt {3}\right )\right ) 2^{{1}/{3}}}{2 {\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{1}/{3}} c_1} \\
\end{align*}
✓ Mathematica. Time used: 0.159 (sec). Leaf size: 46
ode=(3*x^2*y[x]+2*x*y[x]+y[x]^3) + (x^2+y[x]^2)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {K[1]^2+1}{K[1] \left (K[1]^2+3\right )}dK[1]=-x-\log (x)-1+c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(3*x**2*y(x) + 2*x*y(x) + (x**2 + y(x)**2)*Derivative(y(x), x) + y(x)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out