34.2.5 problem 5
Internal
problem
ID
[6035]
Book
:
A
treatise
on
ordinary
and
partial
differential
equations
by
William
Woolsey
Johnson.
1913
Section
:
Chapter
2,
Equations
of
the
first
order
and
degree.
page
20
Problem
number
:
5
Date
solved
:
Sunday, March 30, 2025 at 10:36:36 AM
CAS
classification
:
[_separable]
\begin{align*} \frac {x}{1+y}&=\frac {y y^{\prime }}{1+x} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 494
ode:=x/(1+y(x)) = y(x)/(1+x)*diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-1+4 x^{3}+6 x^{2}+12 c_1 +2 \sqrt {4 x^{6}+12 x^{5}+24 c_1 \,x^{3}+9 x^{4}+36 c_1 \,x^{2}-2 x^{3}+36 c_1^{2}-3 x^{2}-6 c_1}\right )^{{1}/{3}}}{2}+\frac {1}{2 \left (-1+4 x^{3}+6 x^{2}+12 c_1 +2 \sqrt {4 x^{6}+12 x^{5}+24 c_1 \,x^{3}+9 x^{4}+36 c_1 \,x^{2}-2 x^{3}+36 c_1^{2}-3 x^{2}-6 c_1}\right )^{{1}/{3}}}-\frac {1}{2} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (-1+4 x^{3}+6 x^{2}+12 c_1 +2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_1 \right ) \left (2 x^{3}+3 x^{2}+6 c_1 -1\right )}\right )^{{2}/{3}}-i \sqrt {3}+2 \left (-1+4 x^{3}+6 x^{2}+12 c_1 +2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_1 \right ) \left (2 x^{3}+3 x^{2}+6 c_1 -1\right )}\right )^{{1}/{3}}+1}{4 \left (-1+4 x^{3}+6 x^{2}+12 c_1 +2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_1 \right ) \left (2 x^{3}+3 x^{2}+6 c_1 -1\right )}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (-1+4 x^{3}+6 x^{2}+12 c_1 +2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_1 \right ) \left (2 x^{3}+3 x^{2}+6 c_1 -1\right )}\right )^{{2}/{3}}-i \sqrt {3}-2 \left (-1+4 x^{3}+6 x^{2}+12 c_1 +2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_1 \right ) \left (2 x^{3}+3 x^{2}+6 c_1 -1\right )}\right )^{{1}/{3}}-1}{4 \left (-1+4 x^{3}+6 x^{2}+12 c_1 +2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_1 \right ) \left (2 x^{3}+3 x^{2}+6 c_1 -1\right )}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 4.402 (sec). Leaf size: 346
ode=x/(1+y[x])==y[x]/(1+x)*D[y[x],x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{2} \left (\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}+\frac {1}{\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}}-1\right ) \\
y(x)\to \frac {1}{8} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}+\frac {-2-2 i \sqrt {3}}{\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}}-4\right ) \\
y(x)\to \frac {1}{8} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}+\frac {2 i \left (\sqrt {3}+i\right )}{\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}}-4\right ) \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x/(y(x) + 1) - y(x)*Derivative(y(x), x)/(x + 1),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
ZeroDivisionError : polynomial division