21.2.6 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, September 30, 2025 at 07:16:57 AM
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: 0.170 (sec). Leaf size: 73

ode:=y(x)^2*diff(y(x),x) = 2+3*y(x)^6; 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\begin{align*} y &= \frac {3^{{5}/{6}} 2^{{1}/{6}} \tan \left (3 \sqrt {6}\, x \right )^{{1}/{3}}}{3} \\ y &= -\frac {\left (3^{{2}/{3}}-3 i 3^{{1}/{6}}\right ) \tan \left (3 \sqrt {6}\, x \right )^{{1}/{3}} 6^{{1}/{6}}}{6} \\ y &= -\frac {\tan \left (3 \sqrt {6}\, x \right )^{{1}/{3}} \left (3^{{2}/{3}}+3 i 3^{{1}/{6}}\right ) 6^{{1}/{6}}}{6} \\ \end{align*}

✓ Mathematica. Time used: 0.01 (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: 3.662 (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)
 

\[ \left [ y{\left (x \right )} = \frac {\sqrt [6]{2} \cdot 3^{\frac {5}{6}} \sqrt [3]{\tan {\left (3 \sqrt {6} x \right )}}}{3}, \ y{\left (x \right )} = \frac {\sqrt [6]{2} \left (- 3^{\frac {5}{6}} - 3 \sqrt [3]{3} i\right ) \sqrt [3]{\tan {\left (3 \sqrt {6} x \right )}}}{6}, \ y{\left (x \right )} = \frac {\sqrt [6]{2} \left (- 3^{\frac {5}{6}} + 3 \sqrt [3]{3} i\right ) \sqrt [3]{\tan {\left (3 \sqrt {6} x \right )}}}{6}\right ] \]