9.20.2 problem 2

Internal problem ID [3310]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 38, page 173
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 06:35:51 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} x&=y-{y^{\prime }}^{3} \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 211

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

\begin{align*} x -\frac {3 \left (y-x \right )^{{2}/{3}}}{2}-3 \left (y-x \right )^{{1}/{3}}-3 \ln \left (\left (y-x \right )^{{1}/{3}}-1\right )-c_1 &= 0 \\ x +\frac {3 \left (y-x \right )^{{2}/{3}}}{4}-\frac {3 i \sqrt {3}\, \left (y-x \right )^{{2}/{3}}}{4}+\frac {3 \left (y-x \right )^{{1}/{3}}}{2}+\frac {3 i \sqrt {3}\, \left (y-x \right )^{{1}/{3}}}{2}+6 \ln \left (2\right )-3 \ln \left (-4-2 i \sqrt {3}\, \left (y-x \right )^{{1}/{3}}-2 \left (y-x \right )^{{1}/{3}}\right )-c_1 &= 0 \\ x +\frac {3 \left (y-x \right )^{{2}/{3}}}{4}+\frac {3 i \sqrt {3}\, \left (y-x \right )^{{2}/{3}}}{4}+\frac {3 \left (y-x \right )^{{1}/{3}}}{2}-\frac {3 i \sqrt {3}\, \left (y-x \right )^{{1}/{3}}}{2}+6 \ln \left (2\right )-3 \ln \left (2 i \sqrt {3}\, \left (y-x \right )^{{1}/{3}}-2 \left (y-x \right )^{{1}/{3}}-4\right )-c_1 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 7.407 (sec). Leaf size: 330

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

\begin{align*} \text {Solve}\left [\frac {3}{2} (y(x)-x)^{2/3}+3 \sqrt [3]{y(x)-x}+3 \log \left (\sqrt [3]{y(x)-x}-1\right )-x=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {y(x)}{2}+\frac {1}{4} \left (-\frac {1}{2} \sqrt [3]{y(x)-x} \left (4 (y(x)-x)^{2/3}+3 i \sqrt {3} \sqrt [3]{y(x)-x}-3 \sqrt [3]{y(x)-x}-6 i \sqrt {3}-6\right )-6 \log \left (2 i \sqrt [3]{y(x)-x}+\sqrt {2+2 i \sqrt {3}}\right )\right )=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {1}{2} \left (\frac {2 i \left (\sqrt {3}+i\right ) y(x)+3 \sqrt [3]{y(x)-x} \left (2 \sqrt [3]{y(x)-x}-\sqrt {6+6 i \sqrt {3}}-i \sqrt {2+2 i \sqrt {3}}\right )+\left (2-2 i \sqrt {3}\right ) x}{\sqrt {3}+i}+6 i \log \left (\sqrt {2-2 i \sqrt {3}}-2 i \sqrt [3]{y(x)-x}\right )\right )-i (y(x)-x)=c_1,y(x)\right ] \end{align*}

✓ Sympy. Time used: 22.899 (sec). Leaf size: 209

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

\[ \left [ C_{1} + x - \frac {3 \left (- x + y{\left (x \right )}\right )^{\frac {2}{3}}}{2} - 3 \sqrt [3]{- x + y{\left (x \right )}} - 3 \log {\left (1 - \sqrt [3]{- x + y{\left (x \right )}} \right )} = 0, \ C_{1} + x + \frac {3 \left (1 - \sqrt {3} i\right ) \left (- x + y{\left (x \right )}\right )^{\frac {2}{3}}}{4} + \frac {3 \sqrt [3]{- x + y{\left (x \right )}}}{2} + \frac {3 \sqrt {3} i \sqrt [3]{- x + y{\left (x \right )}}}{2} - 3 \log {\left (\sqrt {3} \sqrt [3]{- x + y{\left (x \right )}} - i \sqrt [3]{- x + y{\left (x \right )}} - 2 i \right )} = 0, \ C_{1} + x + \frac {3 \left (1 + \sqrt {3} i\right ) \left (- x + y{\left (x \right )}\right )^{\frac {2}{3}}}{4} + \frac {3 \sqrt [3]{- x + y{\left (x \right )}}}{2} - \frac {3 \sqrt {3} i \sqrt [3]{- x + y{\left (x \right )}}}{2} - 3 \log {\left (\sqrt {3} \sqrt [3]{- x + y{\left (x \right )}} + i \sqrt [3]{- x + y{\left (x \right )}} + 2 i \right )} = 0\right ] \]