76.1.30 problem 30

Internal problem ID [17250]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.1 (Separable equations). Problems at page 44
Problem number : 30
Date solved : Thursday, March 13, 2025 at 09:21:39 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {2 x^{2}}{2 y^{2}-6} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0 \end{align*}

✓ Maple. Time used: 0.124 (sec). Leaf size: 67

ode:=diff(y(x),x) = 2*x^2/(2*y(x)^2-6); 
ic:=y(1) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = -\frac {\left (1+i \sqrt {3}\right ) \left (-4+4 x^{3}+4 \sqrt {x^{6}-2 x^{3}-107}\right )^{{2}/{3}}-12 i \sqrt {3}+12}{4 \left (-4+4 x^{3}+4 \sqrt {x^{6}-2 x^{3}-107}\right )^{{1}/{3}}} \]

✓ Mathematica. Time used: 2.594 (sec). Leaf size: 115

ode=D[y[x],x]==2*x^2/(2*y[x]^2-6); 
ic={y[1]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {-i \sqrt [3]{2} \sqrt {3} \left (x^3+\sqrt {x^6-2 x^3-107}-1\right )^{2/3}-\sqrt [3]{2} \left (x^3+\sqrt {x^6-2 x^3-107}-1\right )^{2/3}+6 i \sqrt {3}-6}{2\ 2^{2/3} \sqrt [3]{x^3+\sqrt {x^6-2 x^3-107}-1}} \]

✓ Sympy. Time used: 47.523 (sec). Leaf size: 105

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x**2/(2*y(x)**2 - 6) + Derivative(y(x), x),0) 
ics = {y(1): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \sqrt [3]{2} \left (\frac {\sqrt [3]{2} \sqrt [3]{- x^{3} + \sqrt {x^{6} - 2 x^{3} - 107} + 1}}{4} + \frac {\sqrt [3]{2} \sqrt {3} i \sqrt [3]{- x^{3} + \sqrt {x^{6} - 2 x^{3} - 107} + 1}}{4} - \frac {6}{\left (-1 - \sqrt {3} i\right ) \sqrt [3]{- x^{3} + \sqrt {x^{6} - 2 x^{3} - 107} + 1}}\right ) \]