76.4.3 problem 3
Internal
problem
ID
[17326]
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
:
3
Date
solved
:
Monday, March 31, 2025 at 03:52:55 PM
CAS
classification
:
[_exact, _rational]
\begin{align*} 3 x^{2}-2 x y+2+\left (6 y^{2}-x^{2}+3\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 368
ode:=3*x^2-2*x*y(x)+2+(6*y(x)^2-x^2+3)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-54 x^{3}-54 c_1 -108 x +6 \sqrt {75 x^{6}+162 c_1 \,x^{3}+378 x^{4}+81 c_1^{2}+324 c_1 x +162 x^{2}+162}\right )^{{2}/{3}}+6 x^{2}-18}{6 \left (-54 x^{3}-54 c_1 -108 x +6 \sqrt {75 x^{6}+162 c_1 \,x^{3}+378 x^{4}+81 c_1^{2}+324 c_1 x +162 x^{2}+162}\right )^{{1}/{3}}} \\
y &= \frac {\left (-i \sqrt {3}-1\right ) \left (-54 x^{3}-54 c_1 -108 x +6 \sqrt {75 x^{6}+162 c_1 \,x^{3}+378 x^{4}+81 c_1^{2}+324 c_1 x +162 x^{2}+162}\right )^{{1}/{3}}}{12}+\frac {\left (i \sqrt {3}-1\right ) \left (x^{2}-3\right )}{2 \left (-54 x^{3}-54 c_1 -108 x +6 \sqrt {75 x^{6}+162 c_1 \,x^{3}+378 x^{4}+81 c_1^{2}+324 c_1 x +162 x^{2}+162}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-54 x^{3}-54 c_1 -108 x +6 \sqrt {75 x^{6}+162 c_1 \,x^{3}+378 x^{4}+81 c_1^{2}+324 c_1 x +162 x^{2}+162}\right )^{{2}/{3}}}{6}+\left (-i \sqrt {3}-1\right ) \left (x^{2}-3\right )}{2 \left (-54 x^{3}-54 c_1 -108 x +6 \sqrt {75 x^{6}+162 c_1 \,x^{3}+378 x^{4}+81 c_1^{2}+324 c_1 x +162 x^{2}+162}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 9.751 (sec). Leaf size: 421
ode=(3*x^2-2*x*y[x]+2)+(6*y[x]^2-x^2+3)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {x^2-3}{\sqrt [3]{6} \sqrt [3]{9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1}}-\frac {\sqrt [3]{9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1}}{6^{2/3}} \\
y(x)\to \frac {\sqrt [3]{6} \left (1+i \sqrt {3}\right ) \left (x^2-3\right )+\left (1-i \sqrt {3}\right ) \left (9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1\right ){}^{2/3}}{2\ 6^{2/3} \sqrt [3]{9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1}} \\
y(x)\to \frac {\sqrt [3]{6} \left (1-i \sqrt {3}\right ) \left (x^2-3\right )+\left (1+i \sqrt {3}\right ) \left (9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1\right ){}^{2/3}}{2\ 6^{2/3} \sqrt [3]{9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(3*x**2 - 2*x*y(x) + (-x**2 + 6*y(x)**2 + 3)*Derivative(y(x), x) + 2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out