23.2.292 problem 311
Internal
problem
ID
[5647]
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
:
311
Date
solved
:
Tuesday, September 30, 2025 at 01:24:04 PM
CAS
classification
:
[_quadrature]
\begin{align*} 4 {y^{\prime }}^{3}+4 y^{\prime }&=x \end{align*}
✓ Maple. Time used: 0.033 (sec). Leaf size: 179
ode:=4*diff(y(x),x)^3+4*diff(y(x),x) = x;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\int \frac {\left (i \sqrt {3}-1\right ) \left (27 x +3 \sqrt {81 x^{2}+192}\right )^{{2}/{3}}+12 i \sqrt {3}+12}{\left (27 x +3 \sqrt {81 x^{2}+192}\right )^{{1}/{3}}}d x}{12}+c_1 \\
y &= -\frac {\int \frac {i \sqrt {3}\, \left (27 x +3 \sqrt {81 x^{2}+192}\right )^{{2}/{3}}+12 i \sqrt {3}+\left (27 x +3 \sqrt {81 x^{2}+192}\right )^{{2}/{3}}-12}{\left (27 x +3 \sqrt {81 x^{2}+192}\right )^{{1}/{3}}}d x}{12}+c_1 \\
y &= \frac {\int \frac {\left (27 x +3 \sqrt {81 x^{2}+192}\right )^{{2}/{3}}-12}{\left (27 x +3 \sqrt {81 x^{2}+192}\right )^{{1}/{3}}}d x}{6}+c_1 \\
\end{align*}
✓ Mathematica. Time used: 2.644 (sec). Leaf size: 571
ode=4 (D[y[x],x])^3 +4 D[y[x],x]==x;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {i \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (27 x^2-32\right ) \left (\sqrt {81 x^2+192}-9 x\right )^{2/3}+3\ 3^{5/6} \left (1-i \sqrt {3}\right ) x \sqrt {27 x^2+64} \left (\sqrt {81 x^2+192}-9 x\right )^{2/3}+8\ 3^{2/3} \left (1+i \sqrt {3}\right ) \left (27 x^2+16\right )-72 \sqrt [6]{3} \left (1+i \sqrt {3}\right ) x \sqrt {27 x^2+64}+48 c_1 \left (\sqrt {81 x^2+192}-9 x\right )^{4/3}}{48 \left (\sqrt {81 x^2+192}-9 x\right )^{4/3}}\\ y(x)&\to \frac {\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \left (32-27 x^2\right ) \left (\sqrt {81 x^2+192}-9 x\right )^{2/3}+3\ 3^{5/6} \left (1+i \sqrt {3}\right ) x \sqrt {27 x^2+64} \left (\sqrt {81 x^2+192}-9 x\right )^{2/3}+8\ 3^{2/3} \left (1-i \sqrt {3}\right ) \left (27 x^2+16\right )+72 i \sqrt [6]{3} \left (\sqrt {3}+i\right ) x \sqrt {27 x^2+64}+48 c_1 \left (\sqrt {81 x^2+192}-9 x\right )^{4/3}}{48 \left (\sqrt {81 x^2+192}-9 x\right )^{4/3}}\\ y(x)&\to \frac {\left (\sqrt [3]{3} \left (\sqrt {81 x^2+192}-9 x\right )^{2/3}-4\ 3^{2/3}\right ) \left (81 x^2-9 \left (\sqrt {81 x^2+192}-2 \sqrt [3]{3} \sqrt [3]{\sqrt {81 x^2+192}-9 x}\right ) x+8\ 3^{2/3} \left (\sqrt {81 x^2+192}-9 x\right )^{2/3}-2\ 3^{5/6} \sqrt {27 x^2+64} \sqrt [3]{\sqrt {81 x^2+192}-9 x}\right )}{72 \left (\sqrt {81 x^2+192}-9 x\right )^{4/3}}+c_1 \end{align*}
✓ Sympy. Time used: 9.716 (sec). Leaf size: 280
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x + 4*Derivative(y(x), x)**3 + 4*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} + \frac {i \left (8 \cdot 3^{\frac {2}{3}} \int \frac {1}{\sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} + 64}}}\, dx + \sqrt [3]{3} \int \sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} + 64}}\, dx - 3^{\frac {5}{6}} i \int \sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} + 64}}\, dx\right )}{6 \left (\sqrt {3} - i\right )}, \ y{\left (x \right )} = C_{1} - \frac {i \left (8 \cdot 3^{\frac {2}{3}} \int \frac {1}{\sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} + 64}}}\, dx + \sqrt [3]{3} \int \sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} + 64}}\, dx + 3^{\frac {5}{6}} i \int \sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} + 64}}\, dx\right )}{6 \left (\sqrt {3} + i\right )}, \ y{\left (x \right )} = C_{1} - \frac {\sqrt [3]{3} \left (- 4 \sqrt [3]{3} \int \frac {1}{\sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} + 64}}}\, dx + \int \sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} + 64}}\, dx\right )}{6}\right ]
\]