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 : Monday, January 27, 2025 at 09:00:55 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*}

✓ Solution by Maple

Time used: 0.218 (sec). Leaf size: 75

dsolve([y(x)^2*diff(y(x),x)=2+3*y(x)^6,y(0) = 0],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*}

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 87

DSolve[{y[x]^2*D[y[x],x]==2+3*y[x]^6,y[0]==0},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*}