80.5.2 problem 3

Internal problem ID [18481]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 32. Problems at page 89
Problem number : 3
Date solved : Thursday, March 13, 2025 at 12:06:22 PM
CAS classification : [_quadrature]

\begin{align*} 2 {y^{\prime }}^{3}+{y^{\prime }}^{2}-y&=0 \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 385

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

\begin{align*} y \left (x \right ) &= 0 \\ -6 \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {\left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}}{3^{{2}/{3}}-\sqrt {3}\, \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}+3^{{1}/{3}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{2}/{3}}}d \textit {\_a} \right )+x -c_{1} &= 0 \\ \frac {-72 \left (\int _{}^{y \left (x \right )}\frac {\left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}}{\left (i 3^{{5}/{6}}+3^{{1}/{3}}-2 \,3^{{1}/{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}\right ) \left (3^{{1}/{3}}+3^{{1}/{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}\right )}d \textit {\_a} \right )+\left (-c_{1} +x \right ) \sqrt {3}+3 i x -3 i c_{1}}{\sqrt {3}+3 i} &= 0 \\ \frac {72 \left (\int _{}^{y \left (x \right )}\frac {\left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}}{\left (-i 3^{{5}/{6}}+3^{{1}/{3}}-2 \,3^{{1}/{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}\right ) \left (3^{{1}/{3}}+3^{{1}/{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}\right )}d \textit {\_a} \right )+\left (-x +c_{1} \right ) \sqrt {3}+3 i x -3 i c_{1}}{-\sqrt {3}+3 i} &= 0 \\ \end{align*}

✗ Mathematica

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

Timed out

✓ Sympy. Time used: 63.755 (sec). Leaf size: 343

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

\[ \left [ - 6 \left (\sqrt {3} - i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1}}{\left (\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} - 1\right ) \left (\sqrt {3} \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} + i \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} + 2 i\right )}\, dy = C_{1} - x, \ - 6 \left (\sqrt {3} + i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1}}{\left (\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} - 1\right ) \left (\sqrt {3} \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} - i \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} - 2 i\right )}\, dy = C_{1} - x, \ \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1}}{\left (- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1\right )^{\frac {2}{3}} + \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} + 1}\, dy = C_{1} - \frac {x}{6}\right ] \]