81.3.24 problem 24
Internal
problem
ID
[18563]
Book
:
A
short
course
on
differential
equations.
By
Donald
Francis
Campbell.
Maxmillan
company.
London.
1907
Section
:
Chapter
III.
Ordinary
differential
equations
of
the
first
order
and
first
degree.
Exercises
at
page
33
Problem
number
:
24
Date
solved
:
Thursday, March 13, 2025 at 12:16:30 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} \left (x^{2}+3 x y-y^{2}\right ) y^{\prime }-3 y^{2}&=0 \end{align*}
✓ Maple. Time used: 0.717 (sec). Leaf size: 98
ode:=(x^2+3*x*y(x)-y(x)^2)*diff(y(x),x)-3*y(x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\[
y \left (x \right ) = \frac {x \left (-\operatorname {RootOf}\left (8 \textit {\_Z}^{10}-12 \textit {\_Z}^{8}+6 \textit {\_Z}^{6}+\left (c_{1} x^{2}-1\right ) \textit {\_Z}^{4}-2 x^{2} c_{1} \textit {\_Z}^{2}+c_{1} x^{2}\right )^{2}+1\right )}{\operatorname {RootOf}\left (8 \textit {\_Z}^{10}-12 \textit {\_Z}^{8}+6 \textit {\_Z}^{6}+\left (c_{1} x^{2}-1\right ) \textit {\_Z}^{4}-2 x^{2} c_{1} \textit {\_Z}^{2}+c_{1} x^{2}\right )^{2}}
\]
✓ Mathematica. Time used: 3.866 (sec). Leaf size: 396
ode=(x^2+3*x*y[x]-y[x]^2)*D[y[x],x]-(3*y[x]^2)==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {Root}\left [\text {$\#$1}^5+3 \text {$\#$1}^4 x+\text {$\#$1}^3 \left (3 x^2+e^{2 c_1}\right )+\text {$\#$1}^2 \left (x^3-3 e^{2 c_1} x\right )+3 \text {$\#$1} e^{2 c_1} x^2-e^{2 c_1} x^3\&,1\right ] \\
y(x)\to \text {Root}\left [\text {$\#$1}^5+3 \text {$\#$1}^4 x+\text {$\#$1}^3 \left (3 x^2+e^{2 c_1}\right )+\text {$\#$1}^2 \left (x^3-3 e^{2 c_1} x\right )+3 \text {$\#$1} e^{2 c_1} x^2-e^{2 c_1} x^3\&,2\right ] \\
y(x)\to \text {Root}\left [\text {$\#$1}^5+3 \text {$\#$1}^4 x+\text {$\#$1}^3 \left (3 x^2+e^{2 c_1}\right )+\text {$\#$1}^2 \left (x^3-3 e^{2 c_1} x\right )+3 \text {$\#$1} e^{2 c_1} x^2-e^{2 c_1} x^3\&,3\right ] \\
y(x)\to \text {Root}\left [\text {$\#$1}^5+3 \text {$\#$1}^4 x+\text {$\#$1}^3 \left (3 x^2+e^{2 c_1}\right )+\text {$\#$1}^2 \left (x^3-3 e^{2 c_1} x\right )+3 \text {$\#$1} e^{2 c_1} x^2-e^{2 c_1} x^3\&,4\right ] \\
y(x)\to \text {Root}\left [\text {$\#$1}^5+3 \text {$\#$1}^4 x+\text {$\#$1}^3 \left (3 x^2+e^{2 c_1}\right )+\text {$\#$1}^2 \left (x^3-3 e^{2 c_1} x\right )+3 \text {$\#$1} e^{2 c_1} x^2-e^{2 c_1} x^3\&,5\right ] \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 0.799 (sec). Leaf size: 27
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((x**2 + 3*x*y(x) - y(x)**2)*Derivative(y(x), x) - 3*y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\log {\left (y{\left (x \right )} \right )} = C_{1} + \log {\left (\frac {\left (\frac {x}{y{\left (x \right )}} - 1\right )^{\frac {3}{2}}}{\left (\frac {x}{y{\left (x \right )}} + 1\right )^{\frac {3}{2}}} \right )}
\]