89.6.36 problem 37
Internal
problem
ID
[24419]
Book
:
A
short
course
in
Differential
Equations.
Earl
D.
Rainville.
Second
edition.
1958.
Macmillan
Publisher,
NY.
CAT
58-5010
Section
:
Chapter
2.
Equations
of
the
first
order
and
first
degree.
Miscellaneous
Exercises
at
page
45
Problem
number
:
37
Date
solved
:
Thursday, October 02, 2025 at 10:26:51 PM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} x^{2}-2 y x -y^{2}-\left (x^{2}+2 y x -y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.013 (sec). Leaf size: 437
ode:=x^2-2*x*y(x)-y(x)^2-(x^2+2*x*y(x)-y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\frac {\left (16 x^{3} c_1^{3}+4+4 \sqrt {-16 x^{6} c_1^{6}+8 x^{3} c_1^{3}+1}\right )^{{1}/{3}}}{2}+\frac {4 x^{2} c_1^{2}}{\left (16 x^{3} c_1^{3}+4+4 \sqrt {-16 x^{6} c_1^{6}+8 x^{3} c_1^{3}+1}\right )^{{1}/{3}}}+c_1 x}{c_1} \\
y &= \frac {-\frac {\left (16 x^{3} c_1^{3}+4+4 \sqrt {-16 x^{6} c_1^{6}+8 x^{3} c_1^{3}+1}\right )^{{1}/{3}}}{4}-\frac {2 x^{2} c_1^{2}}{\left (16 x^{3} c_1^{3}+4+4 \sqrt {-16 x^{6} c_1^{6}+8 x^{3} c_1^{3}+1}\right )^{{1}/{3}}}+c_1 x -\frac {i \sqrt {3}\, \left (-8 x^{2} c_1^{2}+\left (16 x^{3} c_1^{3}+4+4 \sqrt {-16 x^{6} c_1^{6}+8 x^{3} c_1^{3}+1}\right )^{{2}/{3}}\right )}{4 \left (16 x^{3} c_1^{3}+4+4 \sqrt {-16 x^{6} c_1^{6}+8 x^{3} c_1^{3}+1}\right )^{{1}/{3}}}}{c_1} \\
y &= -\frac {8 i \sqrt {3}\, c_1^{2} x^{2}-i \sqrt {3}\, \left (16 x^{3} c_1^{3}+4+4 \sqrt {-16 x^{6} c_1^{6}+8 x^{3} c_1^{3}+1}\right )^{{2}/{3}}+8 x^{2} c_1^{2}-4 c_1 x \left (16 x^{3} c_1^{3}+4+4 \sqrt {-16 x^{6} c_1^{6}+8 x^{3} c_1^{3}+1}\right )^{{1}/{3}}+\left (16 x^{3} c_1^{3}+4+4 \sqrt {-16 x^{6} c_1^{6}+8 x^{3} c_1^{3}+1}\right )^{{2}/{3}}}{4 \left (16 x^{3} c_1^{3}+4+4 \sqrt {-16 x^{6} c_1^{6}+8 x^{3} c_1^{3}+1}\right )^{{1}/{3}} c_1} \\
\end{align*}
✓ Mathematica. Time used: 22.77 (sec). Leaf size: 586
ode=( x^2-2*x*y[x] -y[x]^2 ) - ( x^2+2*x*y[x]-y[x]^2 )*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt [3]{4 x^3+\sqrt {-16 x^6+8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}+\frac {2 \sqrt [3]{2} x^2}{\sqrt [3]{4 x^3+\sqrt {-16 x^6+8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}+x\\ y(x)&\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{4 x^3+\sqrt {-16 x^6+8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{4 x^3+\sqrt {-16 x^6+8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}+x\\ y(x)&\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{4 x^3+\sqrt {-16 x^6+8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}}+\frac {i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{4 x^3+\sqrt {-16 x^6+8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}+x\\ y(x)&\to \sqrt [3]{2} \sqrt [3]{\sqrt {-x^6}+x^3}+\frac {2^{2/3} x^2}{\sqrt [3]{\sqrt {-x^6}+x^3}}+x\\ y(x)&\to \frac {\left (-1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {-x^6}+x^3}}{2^{2/3}}+\frac {i \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{2} \sqrt [3]{\sqrt {-x^6}+x^3}}+x\\ y(x)&\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {-x^6}+x^3}}{2^{2/3}}+\frac {\left (-1-i \sqrt {3}\right ) x^2}{\sqrt [3]{2} \sqrt [3]{\sqrt {-x^6}+x^3}}+x \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2 - 2*x*y(x) - (x**2 + 2*x*y(x) - y(x)**2)*Derivative(y(x), x) - y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out