23.2.257 problem 272
Internal
problem
ID
[5612]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
2.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
SECOND
OR
HIGHER
DEGREE,
page
278
Problem
number
:
272
Date
solved
:
Tuesday, September 30, 2025 at 01:12:03 PM
CAS
classification
:
[[_homogeneous, `class C`], _dAlembert]
\begin{align*} {y^{\prime }}^{3}+x -y&=0 \end{align*}
✓ Maple. Time used: 0.035 (sec). Leaf size: 211
ode:=diff(y(x),x)^3+x-y(x) = 0;
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: 4.73 (sec). Leaf size: 330
ode=(D[y[x],x])^3 +x-y[x]==0 x;
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: 21.786 (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)
\[
\left [ C_{1} + x - \frac {3 \left (- x + y{\left (x \right )}\right )^{\frac {2}{3}}}{2} - 3 \sqrt [3]{- x + y{\left (x \right )}} - 3 \log {\left (1 - \sqrt [3]{- x + y{\left (x \right )}} \right )} = 0, \ C_{1} + x + \frac {3 \left (1 - \sqrt {3} i\right ) \left (- x + y{\left (x \right )}\right )^{\frac {2}{3}}}{4} + \frac {3 \sqrt [3]{- x + y{\left (x \right )}}}{2} + \frac {3 \sqrt {3} i \sqrt [3]{- x + y{\left (x \right )}}}{2} - 3 \log {\left (\sqrt {3} \sqrt [3]{- x + y{\left (x \right )}} - i \sqrt [3]{- x + y{\left (x \right )}} - 2 i \right )} = 0, \ C_{1} + x + \frac {3 \left (1 + \sqrt {3} i\right ) \left (- x + y{\left (x \right )}\right )^{\frac {2}{3}}}{4} + \frac {3 \sqrt [3]{- x + y{\left (x \right )}}}{2} - \frac {3 \sqrt {3} i \sqrt [3]{- x + y{\left (x \right )}}}{2} - 3 \log {\left (\sqrt {3} \sqrt [3]{- x + y{\left (x \right )}} + i \sqrt [3]{- x + y{\left (x \right )}} + 2 i \right )} = 0\right ]
\]