40.3.35 problem 26 (i)
Internal
problem
ID
[6639]
Book
:
Schaums
Outline.
Theory
and
problems
of
Differential
Equations,
1st
edition.
Frank
Ayres.
McGraw
Hill
1952
Section
:
Chapter
5.
Equations
of
first
order
and
first
degree
(Exact
equations).
Supplemetary
problems.
Page
33
Problem
number
:
26
(i)
Date
solved
:
Sunday, March 30, 2025 at 11:13:06 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y \left (y^{2}-2 x^{2}\right )+x \left (2 y^{2}-x^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.131 (sec). Leaf size: 71
ode:=y(x)*(y(x)^2-2*x^2)+x*(2*y(x)^2-x^2)*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: 14.401 (sec). Leaf size: 277
ode=y[x]*(y[x]^2-2*x^2)+x*(2*y[x]^2-x^2)*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: 5.599 (sec). Leaf size: 105
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(-x**2 + 2*y(x)**2)*Derivative(y(x), x) + (-2*x**2 + y(x)**2)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {x^{2} - \frac {\sqrt {C_{1} + x^{6}}}{x}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {x^{2} - \frac {\sqrt {C_{1} + x^{6}}}{x}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {x^{2} + \frac {\sqrt {C_{1} + x^{6}}}{x}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {x^{2} + \frac {\sqrt {C_{1} + x^{6}}}{x}}}{2}\right ]
\]