77.1.50 problem 69 (page 112)
Internal
problem
ID
[17869]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
69
(page
112)
Date
solved
:
Monday, March 31, 2025 at 04:38:27 PM
CAS
classification
:
[_quadrature]
\begin{align*} x {y^{\prime }}^{3}&=1+y^{\prime } \end{align*}
✓ Maple. Time used: 0.030 (sec). Leaf size: 245
ode:=x*diff(y(x),x)^3 = 1+diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {2^{{2}/{3}} 3^{{1}/{3}} \int \frac {\left (1-i \sqrt {3}\right ) \left (\left (\sqrt {3}\, \sqrt {\frac {-4+27 x}{x}}+9\right ) x^{2}\right )^{{2}/{3}}+\left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) 2^{{2}/{3}} x}{\left (\left (\sqrt {3}\, \sqrt {\frac {-4+27 x}{x}}+9\right ) x^{2}\right )^{{1}/{3}} x}d x}{12}+c_1 \\
y &= -\frac {2^{{2}/{3}} 3^{{1}/{3}} \int \frac {\left (1+i \sqrt {3}\right ) \left (\left (\sqrt {3}\, \sqrt {\frac {-4+27 x}{x}}+9\right ) x^{2}\right )^{{2}/{3}}+\left (-i 3^{{5}/{6}}+3^{{1}/{3}}\right ) 2^{{2}/{3}} x}{\left (\left (\sqrt {3}\, \sqrt {\frac {-4+27 x}{x}}+9\right ) x^{2}\right )^{{1}/{3}} x}d x}{12}+c_1 \\
y &= \frac {12^{{1}/{3}} \int \frac {12^{{1}/{3}} x +\left (\left (\sqrt {3}\, \sqrt {\frac {-4+27 x}{x}}+9\right ) x^{2}\right )^{{2}/{3}}}{x \left (\left (\sqrt {3}\, \sqrt {\frac {-4+27 x}{x}}+9\right ) x^{2}\right )^{{1}/{3}}}d x}{6}+c_1 \\
\end{align*}
✓ Mathematica. Time used: 174.18 (sec). Leaf size: 363
ode=x*D[y[x],x]^3==1+D[y[x],x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \int _1^x\frac {2 \sqrt [3]{3} K[1]+\sqrt [3]{2} \left (9 K[1]^2+\sqrt {3} \sqrt {K[1]^3 (27 K[1]-4)}\right )^{2/3}}{6^{2/3} K[1] \sqrt [3]{9 K[1]^2+\sqrt {3} \sqrt {K[1]^3 (27 K[1]-4)}}}dK[1]+c_1 \\
y(x)\to \int _1^x\frac {i \sqrt [3]{3} \left (i+\sqrt {3}\right ) \left (18 K[2]^2+2 \sqrt {3} \sqrt {K[2]^3 (27 K[2]-4)}\right )^{2/3}-2 \sqrt [3]{2} \sqrt [6]{3} \left (3 i+\sqrt {3}\right ) K[2]}{12 K[2] \sqrt [3]{9 K[2]^2+\sqrt {3} \sqrt {K[2]^3 (27 K[2]-4)}}}dK[2]+c_1 \\
y(x)\to \int _1^x-\frac {i \left (\sqrt [3]{2} \sqrt [6]{3} \left (-6-2 i \sqrt {3}\right ) K[3]+\sqrt [3]{3} \left (-i+\sqrt {3}\right ) \left (18 K[3]^2+2 \sqrt {3} \sqrt {K[3]^3 (27 K[3]-4)}\right )^{2/3}\right )}{12 K[3] \sqrt [3]{9 K[3]^2+\sqrt {3} \sqrt {K[3]^3 (27 K[3]-4)}}}dK[3]+c_1 \\
\end{align*}
✓ Sympy. Time used: 30.969 (sec). Leaf size: 320
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*Derivative(y(x), x)**3 - Derivative(y(x), x) - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} + \frac {i \left (- 4 \sqrt [3]{2} \cdot 3^{\frac {2}{3}} \int \frac {1}{x \sqrt [3]{\sqrt {3} \sqrt {\frac {27 - \frac {4}{x}}{x^{2}}} - \frac {9}{x}}}\, dx + \sqrt [3]{12} \int \sqrt [3]{\sqrt {3} \sqrt {\frac {27 - \frac {4}{x}}{x^{2}}} - \frac {9}{x}}\, dx - 2^{\frac {2}{3}} \cdot 3^{\frac {5}{6}} i \int \sqrt [3]{\sqrt {3} \sqrt {\frac {27 - \frac {4}{x}}{x^{2}}} - \frac {9}{x}}\, dx\right )}{6 \left (\sqrt {3} - i\right )}, \ y{\left (x \right )} = C_{1} - \frac {i \left (- 4 \sqrt [3]{2} \cdot 3^{\frac {2}{3}} \int \frac {1}{x \sqrt [3]{\sqrt {3} \sqrt {\frac {27 - \frac {4}{x}}{x^{2}}} - \frac {9}{x}}}\, dx + \sqrt [3]{12} \int \sqrt [3]{\sqrt {3} \sqrt {\frac {27 - \frac {4}{x}}{x^{2}}} - \frac {9}{x}}\, dx + 2^{\frac {2}{3}} \cdot 3^{\frac {5}{6}} i \int \sqrt [3]{\sqrt {3} \sqrt {\frac {27 - \frac {4}{x}}{x^{2}}} - \frac {9}{x}}\, dx\right )}{6 \left (\sqrt {3} + i\right )}, \ y{\left (x \right )} = C_{1} - \frac {\sqrt [3]{18} \int \frac {1}{x \sqrt [3]{\sqrt {3} \sqrt {\frac {27 - \frac {4}{x}}{x^{2}}} - \frac {9}{x}}}\, dx}{3} - \frac {\sqrt [3]{12} \int \sqrt [3]{\sqrt {3} \sqrt {\frac {27 - \frac {4}{x}}{x^{2}}} - \frac {9}{x}}\, dx}{6}\right ]
\]