29.35.13 problem 1046
Internal
problem
ID
[5611]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
35
Problem
number
:
1046
Date
solved
:
Sunday, March 30, 2025 at 09:17:44 AM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3}+\left (1-3 x \right ) {y^{\prime }}^{2}-x \left (1-3 x \right ) y^{\prime }-1-x^{3}&=0 \end{align*}
✓ Maple. Time used: 0.034 (sec). Leaf size: 429
ode:=diff(y(x),x)^3+(1-3*x)*diff(y(x),x)^2-x*(1-3*x)*diff(y(x),x)-1-x^3 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {\int \frac {4+i \left (\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{2}/{3}}+12 x -4\right ) \sqrt {3}-12 \left (1+\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{1}/{3}}\right ) x +\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{2}/{3}}+4 \left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{1}/{3}}}{\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{1}/{3}}}d x}{12}+c_1 \\
y &= \frac {\int \frac {\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{2}/{3}}+12 \left (i \sqrt {3}+\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{1}/{3}}+1\right ) \left (x -\frac {1}{3}\right )}{\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{1}/{3}}}d x}{12}+c_1 \\
y &= \frac {\int \frac {4+6 \left (-2+\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{1}/{3}}\right ) x +\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{2}/{3}}-2 \left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{1}/{3}}}{\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{1}/{3}}}d x}{6}+c_1 \\
\end{align*}
✓ Mathematica. Time used: 94.042 (sec). Leaf size: 1649
ode=(D[y[x],x])^3+(1-3*x)(D[y[x],x])^2-x*(1-3*x)*D[y[x],x]-1 -x^3==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Sympy. Time used: 75.228 (sec). Leaf size: 571
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**3 - x*(1 - 3*x)*Derivative(y(x), x) + (1 - 3*x)*Derivative(y(x), x)**2 + Derivative(y(x), x)**3 - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} + \frac {x^{2}}{2} - \frac {x}{3} + \sqrt [3]{2} \int \frac {x}{\sqrt [3]{- 9 x + 3 \sqrt {3} \sqrt {4 x^{3} - x^{2} + 18 x + 23} - 25}}\, dx - \frac {\sqrt [3]{2} \int \frac {1}{\sqrt [3]{- 9 x + 3 \sqrt {3} \sqrt {4 x^{3} - x^{2} + 18 x + 23} - 25}}\, dx}{3} - \frac {2^{\frac {2}{3}} \int \sqrt [3]{- 9 x + 3 \sqrt {3} \sqrt {4 x^{3} - x^{2} + 18 x + 23} - 25}\, dx}{6}, \ y{\left (x \right )} = C_{1} + \frac {3 \sqrt {3} x^{2} - 3 i x^{2} - 2 \sqrt {3} x + 2 i x + 12 \sqrt [3]{2} i \int \frac {x}{\sqrt [3]{- 9 x + 3 \sqrt {3} \sqrt {4 x^{3} - x^{2} + 18 x + 23} - 25}}\, dx - 4 \sqrt [3]{2} i \int \frac {1}{\sqrt [3]{- 9 x + 3 \sqrt {3} \sqrt {4 x^{3} - x^{2} + 18 x + 23} - 25}}\, dx + 2^{\frac {2}{3}} \sqrt {3} \int \sqrt [3]{- 9 x + 3 \sqrt {3} \sqrt {4 x^{3} - x^{2} + 18 x + 23} - 25}\, dx + 2^{\frac {2}{3}} i \int \sqrt [3]{- 9 x + 3 \sqrt {3} \sqrt {4 x^{3} - x^{2} + 18 x + 23} - 25}\, dx}{6 \left (\sqrt {3} - i\right )}, \ y{\left (x \right )} = C_{1} + \frac {3 \sqrt {3} x^{2} + 3 i x^{2} - 2 \sqrt {3} x - 2 i x - 12 \sqrt [3]{2} i \int \frac {x}{\sqrt [3]{- 9 x + 3 \sqrt {3} \sqrt {4 x^{3} - x^{2} + 18 x + 23} - 25}}\, dx + 4 \sqrt [3]{2} i \int \frac {1}{\sqrt [3]{- 9 x + 3 \sqrt {3} \sqrt {4 x^{3} - x^{2} + 18 x + 23} - 25}}\, dx + 2^{\frac {2}{3}} \sqrt {3} \int \sqrt [3]{- 9 x + 3 \sqrt {3} \sqrt {4 x^{3} - x^{2} + 18 x + 23} - 25}\, dx - 2^{\frac {2}{3}} i \int \sqrt [3]{- 9 x + 3 \sqrt {3} \sqrt {4 x^{3} - x^{2} + 18 x + 23} - 25}\, dx}{6 \left (\sqrt {3} + i\right )}\right ]
\]