60.1.16 problem Problem 16
Internal
problem
ID
[15144]
Book
:
Differential
equations
and
the
calculus
of
variations
by
L.
ElSGOLTS.
MIR
PUBLISHERS,
MOSCOW,
Third
printing
1977.
Section
:
Chapter
1,
First-Order
Differential
Equations.
Problems
page
88
Problem
number
:
Problem
16
Date
solved
:
Thursday, October 02, 2025 at 10:04:28 AM
CAS
classification
:
[_quadrature]
\begin{align*} x&={y^{\prime }}^{3}-y^{\prime }+2 \end{align*}
✓ Maple. Time used: 0.033 (sec). Leaf size: 207
ode:=x = diff(y(x),x)^3-diff(y(x),x)+2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {\int \frac {i \sqrt {3}\, \left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{{2}/{3}}-12 i \sqrt {3}+\left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{{2}/{3}}+12}{\left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{{1}/{3}}}d x}{12}+c_1 \\
y &= \frac {\int \frac {\left (i \sqrt {3}-1\right ) \left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{{2}/{3}}-12 i \sqrt {3}-12}{\left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{{1}/{3}}}d x}{12}+c_1 \\
y &= \frac {\int \frac {\left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{{2}/{3}}+12}{\left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{{1}/{3}}}d x}{6}+c_1 \\
\end{align*}
✓ Mathematica. Time used: 106.375 (sec). Leaf size: 477
ode=x==D[y[x],x]^3-D[y[x],x]+2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {36\ 6^{2/3} \left (\sqrt {81 x^2-324 x+312}-9 x+18\right )^{2/3} (x-2)+72\ 3^{5/6} \sqrt {27 x^2-108 x+104} \sqrt [3]{\sqrt {81 x^2-324 x+312}-9 x+18} (x-2)-12\ 2^{2/3} \sqrt [6]{3} \sqrt {27 x^2-108 x+104} \left (\sqrt {81 x^2-324 x+312}-9 x+18\right )^{2/3}-24 \sqrt [3]{3} \left (27 x^2-108 x+107\right ) \sqrt [3]{\sqrt {81 x^2-324 x+312}-9 x+18}+\sqrt [3]{2} \left (2916 x^3-17496 x^2-12 \left (27 x^2-108 x+109\right ) \sqrt {81 x^2-324 x+312}+34884 x-23112\right )}{24\ 6^{2/3} \left (\sqrt {81 x^2-324 x+312}-9 x+18\right )^{5/3}}+c_1\\ y(x)&\to \int _1^x\frac {\sqrt [3]{2} \left (1-i \sqrt {3}\right ) \left (-9 K[1]+\sqrt {81 K[1]^2-324 K[1]+312}+18\right )^{2/3}+2 \left (\sqrt [3]{3}+i 3^{5/6}\right )}{2\ 6^{2/3} \sqrt [3]{-9 K[1]+\sqrt {81 K[1]^2-324 K[1]+312}+18}}dK[1]+c_1\\ y(x)&\to \int _1^x\frac {\sqrt [3]{2} \left (1+i \sqrt {3}\right ) \left (-9 K[2]+\sqrt {81 K[2]^2-324 K[2]+312}+18\right )^{2/3}+2 \left (\sqrt [3]{3}-i 3^{5/6}\right )}{2\ 6^{2/3} \sqrt [3]{-9 K[2]+\sqrt {81 K[2]^2-324 K[2]+312}+18}}dK[2]+c_1 \end{align*}
✓ Sympy. Time used: 8.403 (sec). Leaf size: 342
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x - Derivative(y(x), x)**3 + Derivative(y(x), x) - 2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} + \frac {i \left (- 4 \sqrt [3]{2} \cdot 3^{\frac {2}{3}} \int \frac {1}{\sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} - 108 x + 104} + 18}}\, dx + \sqrt [3]{12} \int \sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} - 108 x + 104} + 18}\, dx - 2^{\frac {2}{3}} \cdot 3^{\frac {5}{6}} i \int \sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} - 108 x + 104} + 18}\, dx\right )}{6 \left (\sqrt {3} - i\right )}, \ y{\left (x \right )} = C_{1} - \frac {i \left (- 4 \sqrt [3]{2} \cdot 3^{\frac {2}{3}} \int \frac {1}{\sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} - 108 x + 104} + 18}}\, dx + \sqrt [3]{12} \int \sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} - 108 x + 104} + 18}\, dx + 2^{\frac {2}{3}} \cdot 3^{\frac {5}{6}} i \int \sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} - 108 x + 104} + 18}\, dx\right )}{6 \left (\sqrt {3} + i\right )}, \ y{\left (x \right )} = C_{1} - \frac {\sqrt [3]{18} \int \frac {1}{\sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} - 108 x + 104} + 18}}\, dx}{3} - \frac {\sqrt [3]{12} \int \sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} - 108 x + 104} + 18}\, dx}{6}\right ]
\]