26.5.8 problem Exercise 11.8, page 97

Internal problem ID [6981]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 11, Bernoulli Equations
Problem number : Exercise 11.8, page 97
Date solved : Tuesday, September 30, 2025 at 04:07:38 PM
CAS classification : [_rational, _Bernoulli]

\begin{align*} \left (-x^{3}+1\right ) y^{\prime }-2 \left (1+x \right ) y&=y^{{5}/{2}} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 38

ode:=(-x^3+1)*diff(y(x),x)-2*(1+x)*y(x) = y(x)^(5/2); 
dsolve(ode,y(x), singsol=all);
 

\[ -\frac {\left (x -1\right )^{2} c_1}{x^{2}+x +1}+\frac {1}{y^{{3}/{2}}}+\frac {3}{4 x^{2}+4 x +4} = 0 \]

✓ Mathematica. Time used: 10.688 (sec). Leaf size: 100

ode=(1-x^3)*D[y[x],x]-2*(1+x)*y[x]==y[x]^(5/2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {2^{2/3}}{\left (\exp \left (-\frac {3}{2} \int _1^x-\frac {2 (K[1]+1)}{K[1]^3-1}dK[1]\right ) \left (-3 \int _1^x-\frac {\exp \left (\frac {3}{2} \int _1^{K[2]}-\frac {2 (K[1]+1)}{K[1]^3-1}dK[1]\right )}{K[2]^3-1}dK[2]+2 c_1\right )\right ){}^{2/3}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 60.424 (sec). Leaf size: 122

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - x**3)*Derivative(y(x), x) - (2*x + 2)*y(x) - y(x)**(5/2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = 2 \sqrt [3]{2} \left (\frac {x^{2} + x + 1}{4 C_{1} x^{2} - 8 C_{1} x + 4 C_{1} - 3}\right )^{\frac {2}{3}}, \ y{\left (x \right )} = \sqrt [3]{2} \left (\frac {x^{2} + x + 1}{4 C_{1} x^{2} - 8 C_{1} x + 4 C_{1} - 3}\right )^{\frac {2}{3}} \left (-1 - \sqrt {3} i\right ), \ y{\left (x \right )} = \sqrt [3]{2} \left (\frac {x^{2} + x + 1}{4 C_{1} x^{2} - 8 C_{1} x + 4 C_{1} - 3}\right )^{\frac {2}{3}} \left (-1 + \sqrt {3} i\right )\right ] \]