89.28.15 problem 15
Internal
problem
ID
[24903]
Book
:
A
short
course
in
Differential
Equations.
Earl
D.
Rainville.
Second
edition.
1958.
Macmillan
Publisher,
NY.
CAT
58-5010
Section
:
Chapter
16.
Equations
of
order
one
and
higher
degree.
Exercises
at
page
229
Problem
number
:
15
Date
solved
:
Thursday, October 02, 2025 at 10:49:18 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} \left (x^{2}+y^{2}\right )^{2} {y^{\prime }}^{2}&=4 x^{2} y^{2} \end{align*}
✓ Maple. Time used: 0.114 (sec). Leaf size: 255
ode:=(x^2+y(x)^2)^2*diff(y(x),x)^2 = 4*x^2*y(x)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {1-\sqrt {4 c_1^{2} x^{2}+1}}{2 c_1} \\
y &= \frac {1+\sqrt {4 c_1^{2} x^{2}+1}}{2 c_1} \\
y &= -\frac {2 \left (c_1 \,x^{2}-\frac {\left (4+4 \sqrt {4 c_1^{3} x^{6}+1}\right )^{{2}/{3}}}{4}\right )}{\sqrt {c_1}\, \left (4+4 \sqrt {4 c_1^{3} x^{6}+1}\right )^{{1}/{3}}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (4+4 \sqrt {4 c_1^{3} x^{6}+1}\right )^{{1}/{3}}}{4 \sqrt {c_1}}-\frac {\sqrt {c_1}\, \left (i \sqrt {3}-1\right ) x^{2}}{\left (4+4 \sqrt {4 c_1^{3} x^{6}+1}\right )^{{1}/{3}}} \\
y &= \frac {4 i \sqrt {3}\, c_1 \,x^{2}+i \sqrt {3}\, \left (4+4 \sqrt {4 c_1^{3} x^{6}+1}\right )^{{2}/{3}}+4 c_1 \,x^{2}-\left (4+4 \sqrt {4 c_1^{3} x^{6}+1}\right )^{{2}/{3}}}{4 \left (4+4 \sqrt {4 c_1^{3} x^{6}+1}\right )^{{1}/{3}} \sqrt {c_1}} \\
\end{align*}
✓ Mathematica. Time used: 11.638 (sec). Leaf size: 345
ode=(x^2+y[x]^2)^2*D[y[x],x]^2 ==4*x^2*y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{2} \left (-\sqrt {4 x^2+e^{2 c_1}}-e^{c_1}\right )\\ y(x)&\to \frac {1}{2} \left (\sqrt {4 x^2+e^{2 c_1}}-e^{c_1}\right )\\ y(x)&\to \frac {\sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}\\ y(x)&\to \frac {i 2^{2/3} \left (\sqrt {3}+i\right ) \left (\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}+\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x^2}{4 \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}\\ y(x)&\to \frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}}\\ y(x)&\to 0 \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-4*x**2*y(x)**2 + (x**2 + y(x)**2)**2*Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out