82.9.1 problem Ex. 1
Internal
problem
ID
[18682]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
II.
Equations
of
the
first
order
and
of
the
first
degree.
Exercises
at
page
26
Problem
number
:
Ex.
1
Date
solved
:
Monday, March 31, 2025 at 05:56:32 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y^{3}-2 y x^{2}+\left (2 x y^{2}-x^{3}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.125 (sec). Leaf size: 71
ode:=y(x)^3-2*x^2*y(x)+(2*x*y(x)^2-x^3)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\sqrt {\frac {2 c_1 \,x^{3}-2 \sqrt {c_1^{2} x^{6}+4}}{c_1 \,x^{3}}}\, x}{2} \\
y &= \frac {\sqrt {2}\, \sqrt {\frac {c_1 \,x^{3}+\sqrt {c_1^{2} x^{6}+4}}{c_1 \,x^{3}}}\, x}{2} \\
\end{align*}
✓ Mathematica. Time used: 12.028 (sec). Leaf size: 277
ode=(y[x]^3-2*y[x]*x^2)+(2*x*y[x]^2-x^3)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt {x^2-\frac {\sqrt {x^6-4 e^{2 c_1}}}{x}}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {x^2-\frac {\sqrt {x^6-4 e^{2 c_1}}}{x}}}{\sqrt {2}} \\
y(x)\to -\frac {\sqrt {\frac {x^3+\sqrt {x^6-4 e^{2 c_1}}}{x}}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {\frac {x^3+\sqrt {x^6-4 e^{2 c_1}}}{x}}}{\sqrt {2}} \\
y(x)\to -\frac {\sqrt {x^2-\frac {\sqrt {x^6}}{x}}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {x^2-\frac {\sqrt {x^6}}{x}}}{\sqrt {2}} \\
y(x)\to -\frac {\sqrt {\frac {\sqrt {x^6}+x^3}{x}}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {\frac {\sqrt {x^6}+x^3}{x}}}{\sqrt {2}} \\
\end{align*}
✓ Sympy. Time used: 5.577 (sec). Leaf size: 105
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-2*x**2*y(x) + (-x**3 + 2*x*y(x)**2)*Derivative(y(x), x) + y(x)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {x^{2} - \frac {\sqrt {C_{1} + x^{6}}}{x}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {x^{2} - \frac {\sqrt {C_{1} + x^{6}}}{x}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {x^{2} + \frac {\sqrt {C_{1} + x^{6}}}{x}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {x^{2} + \frac {\sqrt {C_{1} + x^{6}}}{x}}}{2}\right ]
\]