83.23.7 problem 7
Internal
problem
ID
[19209]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
V.
Singular
solutions.
Exercise
V
at
page
76
Problem
number
:
7
Date
solved
:
Thursday, March 13, 2025 at 01:55:40 PM
CAS
classification
:
[_quadrature]
\begin{align*} \left (8 {y^{\prime }}^{3}-27\right ) x&=\frac {12 {y^{\prime }}^{2}}{x} \end{align*}
✓ Maple. Time used: 0.037 (sec). Leaf size: 335
ode:=(8*diff(y(x),x)^3-27)*x = 12*diff(y(x),x)^2/x;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= -\frac {\left (\int \frac {\left (i \sqrt {3}\, \left (108 x^{6}+12 \sqrt {3}\, \sqrt {27 x^{6}+4}\, x^{3}+8\right )^{{1}/{3}}+\left (108 x^{6}+12 \sqrt {3}\, \sqrt {27 x^{6}+4}\, x^{3}+8\right )^{{1}/{3}}+2 i \sqrt {3}-2\right ) \left (\left (108 x^{6}+12 \sqrt {3}\, \sqrt {27 x^{6}+4}\, x^{3}+8\right )^{{1}/{3}}-2\right )}{x^{2} \left (108 x^{6}+12 \sqrt {3}\, \sqrt {27 x^{6}+4}\, x^{3}+8\right )^{{1}/{3}}}d x \right )}{8}+c_{1} \\
y \left (x \right ) &= \frac {\left (\int \frac {\left (\left (108 x^{6}+12 \sqrt {3}\, \sqrt {27 x^{6}+4}\, x^{3}+8\right )^{{1}/{3}}-2\right ) \left (\left (i \sqrt {3}-1\right ) \left (108 x^{6}+12 \sqrt {3}\, \sqrt {27 x^{6}+4}\, x^{3}+8\right )^{{1}/{3}}+2 i \sqrt {3}+2\right )}{\left (108 x^{6}+12 \sqrt {3}\, \sqrt {27 x^{6}+4}\, x^{3}+8\right )^{{1}/{3}} x^{2}}d x \right )}{8}+c_{1} \\
y \left (x \right ) &= \frac {\left (\int \frac {\left (108 x^{6}+12 \sqrt {3}\, \sqrt {27 x^{6}+4}\, x^{3}+8\right )^{{2}/{3}}+2 \left (108 x^{6}+12 \sqrt {3}\, \sqrt {27 x^{6}+4}\, x^{3}+8\right )^{{1}/{3}}+4}{x^{2} \left (108 x^{6}+12 \sqrt {3}\, \sqrt {27 x^{6}+4}\, x^{3}+8\right )^{{1}/{3}}}d x \right )}{4}+c_{1} \\
\end{align*}
✓ Mathematica. Time used: 41.471 (sec). Leaf size: 371
ode=(8*D[y[x],x]^3-27)*x==12*D[y[x],x]^2/x;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \int _1^x\frac {2^{2/3} \sqrt [3]{27 K[1]^6+3 \sqrt {3} \sqrt {K[1]^6 \left (27 K[1]^6+4\right )}+2}+\frac {2}{\sqrt [3]{\frac {27 K[1]^6}{2}+\frac {3}{2} \sqrt {3} \sqrt {K[1]^6 \left (27 K[1]^6+4\right )}+1}}+2}{4 K[1]^2}dK[1]+c_1 \\
y(x)\to \int _1^x\frac {i 2^{2/3} \left (i+\sqrt {3}\right ) \sqrt [3]{27 K[2]^6+3 \sqrt {3} \sqrt {K[2]^6 \left (27 K[2]^6+4\right )}+2}-\frac {2 \left (1+i \sqrt {3}\right )}{\sqrt [3]{\frac {27 K[2]^6}{2}+\frac {3}{2} \sqrt {3} \sqrt {K[2]^6 \left (27 K[2]^6+4\right )}+1}}+4}{8 K[2]^2}dK[2]+c_1 \\
y(x)\to \int _1^x\frac {-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{27 K[3]^6+3 \sqrt {3} \sqrt {K[3]^6 \left (27 K[3]^6+4\right )}+2}+\frac {2 i \left (i+\sqrt {3}\right )}{\sqrt [3]{\frac {27 K[3]^6}{2}+\frac {3}{2} \sqrt {3} \sqrt {K[3]^6 \left (27 K[3]^6+4\right )}+1}}+4}{8 K[3]^2}dK[3]+c_1 \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(8*Derivative(y(x), x)**3 - 27) - 12*Derivative(y(x), x)**2/x,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out