32.6.49 problem Exercise 12.49, page 103

Internal problem ID [5914]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous Methods
Problem number : Exercise 12.49, page 103
Date solved : Sunday, March 30, 2025 at 10:27:08 AM
CAS classification : [_separable]

\begin{align*} \left (2 y^{3}+y\right ) y^{\prime }-2 x^{3}-x&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 113

ode:=(2*y(x)^3+y(x))*diff(y(x),x)-2*x^3-x = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {-2-2 \sqrt {4 x^{4}+4 x^{2}+8 c_1 +1}}}{2} \\ y &= \frac {\sqrt {-2-2 \sqrt {4 x^{4}+4 x^{2}+8 c_1 +1}}}{2} \\ y &= -\frac {\sqrt {-2+2 \sqrt {4 x^{4}+4 x^{2}+8 c_1 +1}}}{2} \\ y &= \frac {\sqrt {-2+2 \sqrt {4 x^{4}+4 x^{2}+8 c_1 +1}}}{2} \\ \end{align*}

✓ Mathematica. Time used: 2.427 (sec). Leaf size: 151

ode=(2*y[x]^3+y[x])*D[y[x],x]-2*x^3-x==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\sqrt {-1-\sqrt {4 x^4+4 x^2+1+8 c_1}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-1-\sqrt {4 x^4+4 x^2+1+8 c_1}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {-1+\sqrt {4 x^4+4 x^2+1+8 c_1}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-1+\sqrt {4 x^4+4 x^2+1+8 c_1}}}{\sqrt {2}} \\ \end{align*}

✓ Sympy. Time used: 3.026 (sec). Leaf size: 122

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x**3 - x + (2*y(x)**3 + y(x))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- \sqrt {C_{1} + 4 x^{4} + 4 x^{2}} - 1}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- \sqrt {C_{1} + 4 x^{4} + 4 x^{2}} - 1}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {\sqrt {C_{1} + 4 x^{4} + 4 x^{2}} - 1}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {\sqrt {C_{1} + 4 x^{4} + 4 x^{2}} - 1}}{2}\right ] \]