27.2.6 problem 6
Internal
problem
ID
[4306]
Book
:
An
introduction
to
the
solution
and
applications
of
differential
equations,
J.W.
Searl,
1966
Section
:
Chapter
4,
Ex.
4.2
Problem
number
:
6
Date
solved
:
Tuesday, March 04, 2025 at 06:19:38 PM
CAS
classification
:
[_quadrature]
\begin{align*} y^{2} y^{\prime }&=2+3 y^{6} \end{align*}
With initial conditions
\begin{align*} y \left (0\right )&=0 \end{align*}
✓ Maple. Time used: 0.218 (sec). Leaf size: 75
ode:=y(x)^2*diff(y(x),x) = 2+3*y(x)^6;
ic:=y(0) = 0;
dsolve([ode,ic],y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \frac {3^{{5}/{6}} 2^{{1}/{6}} \tan \left (3 x \sqrt {6}\right )^{{1}/{3}}}{3} \\
y \left (x \right ) &= \frac {\tan \left (3 x \sqrt {6}\right )^{{1}/{3}} \left (3 i 3^{{1}/{6}}-3^{{2}/{3}}\right ) 6^{{1}/{6}}}{6} \\
y \left (x \right ) &= -\frac {\tan \left (3 x \sqrt {6}\right )^{{1}/{3}} \left (3 i 3^{{1}/{6}}+3^{{2}/{3}}\right ) 6^{{1}/{6}}}{6} \\
\end{align*}
✓ Mathematica. Time used: 0.015 (sec). Leaf size: 87
ode=y[x]^2*D[y[x],x]==2+3*y[x]^6;
ic=y[0]==0;
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \sqrt [6]{\frac {2}{3}} \sqrt [3]{\tan \left (3 \sqrt {6} x\right )} \\
y(x)\to -\sqrt [3]{-1} \sqrt [6]{\frac {2}{3}} \sqrt [3]{\tan \left (3 \sqrt {6} x\right )} \\
y(x)\to (-1)^{2/3} \sqrt [6]{\frac {2}{3}} \sqrt [3]{\tan \left (3 \sqrt {6} x\right )} \\
\end{align*}
✓ Sympy. Time used: 5.260 (sec). Leaf size: 100
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-3*y(x)**6 + y(x)**2*Derivative(y(x), x) - 2,0)
ics = {y(0): 0}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\sqrt [6]{2} \cdot 3^{\frac {5}{6}} \sqrt [3]{\tan {\left (3 \sqrt {6} x \right )}}}{3}, \ y{\left (x \right )} = \frac {\sqrt [6]{2} \left (- 3^{\frac {5}{6}} - 3 \sqrt [3]{3} i\right ) \sqrt [3]{\tan {\left (3 \sqrt {6} x \right )}}}{6}, \ y{\left (x \right )} = \frac {\sqrt [6]{2} \left (- 3^{\frac {5}{6}} + 3 \sqrt [3]{3} i\right ) \sqrt [3]{\tan {\left (3 \sqrt {6} x \right )}}}{6}\right ]
\]