1.3.15 problem 15

Internal problem ID [55]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 1. First order differential equations. Section 1.4 (separable equations). Problems at page 43
Problem number : 15
Date solved : Tuesday, September 30, 2025 at 03:41:02 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 840

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 33.88 (sec). Leaf size: 842

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

\begin{align*} y(x)&\to -\frac {\frac {8 \sqrt [3]{2} x^2}{\sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}}+2^{2/3} \sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}+4 x}{6 (x \log (x)+c_1 x+1)}\\ y(x)&\to \frac {\frac {8 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}}+2^{2/3} \left (1-i \sqrt {3}\right ) \sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}-8 x}{12 (x \log (x)+c_1 x+1)}\\ y(x)&\to \frac {\frac {8 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) x^2}{\sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}}+2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}-8 x}{12 (x \log (x)+c_1 x+1)}\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

Timed Out