84.10.6 problem 5.16
Internal
problem
ID
[22134]
Book
:
Schaums
outline
series.
Differential
Equations
By
Richard
Bronson.
1973.
McGraw-Hill
Inc.
ISBN
0-07-008009-7
Section
:
Chapter
5.
Homogeneous
differential
equations.
Supplementary
problems
Problem
number
:
5.16
Date
solved
:
Thursday, October 02, 2025 at 08:29:06 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y^{\prime }&=\frac {2 x y}{y^{2}-x^{2}} \end{align*}
✓ Maple. Time used: 0.012 (sec). Leaf size: 208
ode:=diff(y(x),x) = 2*x*y(x)/(-x^2+y(x)^2);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {4 c_1 \,x^{2}+\left (4+4 \sqrt {-4 x^{6} c_1^{3}+1}\right )^{{2}/{3}}}{2 \left (4+4 \sqrt {-4 x^{6} c_1^{3}+1}\right )^{{1}/{3}} \sqrt {c_1}} \\
y &= \frac {4 i \sqrt {3}\, c_1 \,x^{2}-i \sqrt {3}\, \left (4+4 \sqrt {-4 x^{6} c_1^{3}+1}\right )^{{2}/{3}}-4 c_1 \,x^{2}-\left (4+4 \sqrt {-4 x^{6} c_1^{3}+1}\right )^{{2}/{3}}}{4 \left (4+4 \sqrt {-4 x^{6} c_1^{3}+1}\right )^{{1}/{3}} \sqrt {c_1}} \\
y &= \frac {\left (4+4 \sqrt {-4 x^{6} c_1^{3}+1}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{4 \sqrt {c_1}}-\frac {x^{2} \left (1+i \sqrt {3}\right ) \sqrt {c_1}}{\left (4+4 \sqrt {-4 x^{6} c_1^{3}+1}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 23.639 (sec). Leaf size: 426
ode=D[y[x],x]==(2*x*y[x])/(y[x]^2-x^2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} 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 \left (\sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (\sqrt {-4 x^6+e^{6 c_1}}-e^{3 c_1}\right ){}^{2/3}-2 \left (\sqrt {3}-i\right ) x^2\right )}{2\ 2^{2/3} \sqrt [3]{\sqrt {-4 x^6+e^{6 c_1}}-e^{3 c_1}}}\\ y(x)&\to \frac {2^{2/3} \left (-1-i \sqrt {3}\right ) \left (\sqrt {-4 x^6+e^{6 c_1}}-e^{3 c_1}\right ){}^{2/3}+2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^2}{4 \sqrt [3]{\sqrt {-4 x^6+e^{6 c_1}}-e^{3 c_1}}}\\ y(x)&\to 0\\ y(x)&\to \frac {\sqrt [3]{-x^6}+x^2}{\sqrt [6]{-x^6}}\\ y(x)&\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^6}+\left (-1-i \sqrt {3}\right ) x^2}{2 \sqrt [6]{-x^6}}\\ y(x)&\to \frac {1}{2} \sqrt [6]{-x^6} \left (\frac {\left (1-i \sqrt {3}\right ) \left (-x^6\right )^{2/3}}{x^4}-i \sqrt {3}-1\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-2*x*y(x)/(-x**2 + y(x)**2) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out