83.45.8 problem Ex 8 page 55
Internal
problem
ID
[19480]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Book
Solved
Excercises.
Chapter
IV.
Equations
of
the
first
order
but
not
of
the
first
degree
Problem
number
:
Ex
8
page
55
Date
solved
:
Monday, March 31, 2025 at 07:21:47 PM
CAS
classification
:
[_quadrature]
\begin{align*} x {y^{\prime }}^{3}&=a +b y^{\prime } \end{align*}
✓ Maple. Time used: 0.034 (sec). Leaf size: 305
ode:=x*diff(y(x),x)^3 = a+b*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 {27 a^{2} x -4 b^{3}}{x}}+9 a \right ) x^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) x b}{{\left (\left (\sqrt {3}\, \sqrt {\frac {27 a^{2} x -4 b^{3}}{x}}+9 a \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 {27 a^{2} x -4 b^{3}}{x}}+9 a \right ) x^{2}\right )}^{{2}/{3}}-2^{{2}/{3}} x b \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )}{{\left (\left (\sqrt {3}\, \sqrt {\frac {27 a^{2} x -4 b^{3}}{x}}+9 a \right ) x^{2}\right )}^{{1}/{3}} x}d x}{12}+c_1 \\
y &= \frac {12^{{1}/{3}} \int \frac {b 12^{{1}/{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {27 a^{2} x -4 b^{3}}{x}}+9 a \right ) x^{2}\right )}^{{2}/{3}}}{x {\left (\left (\sqrt {3}\, \sqrt {\frac {27 a^{2} x -4 b^{3}}{x}}+9 a \right ) x^{2}\right )}^{{1}/{3}}}d x}{6}+c_1 \\
\end{align*}
✓ Mathematica. Time used: 166.243 (sec). Leaf size: 535
ode=x*D[y[x],x]^3==a+b*D[y[x],x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt [3]{6} x \left (-9 \sqrt {3} a x \sqrt {x^3 \left (27 a^2 x-4 b^3\right )}+2\ 2^{2/3} \sqrt [3]{3} b \left (\sqrt {3} \sqrt {x^3 \left (27 a^2 x-4 b^3\right )}-9 a x^2\right )^{4/3}+4 \sqrt [3]{2} b^2 x \left (3 \sqrt {3} \sqrt {x^3 \left (27 a^2 x-4 b^3\right )}-27 a x^2\right )^{2/3}+81 a^2 x^3\right )}{\sqrt [3]{\sqrt {3} \sqrt {x^3 \left (27 a^2 x-4 b^3\right )}-9 a x^2} \left (\sqrt [3]{2} \left (\sqrt {3} \sqrt {x^3 \left (27 a^2 x-4 b^3\right )}-9 a x^2\right )^{2/3}+2 \sqrt [3]{3} b x\right )^2}+c_1 \\
y(x)\to \int _1^x\frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (3 i+\sqrt {3}\right ) b K[1]+\sqrt [3]{3} \left (1-i \sqrt {3}\right ) \left (2 \sqrt {3} \sqrt {K[1]^3 \left (27 a^2 K[1]-4 b^3\right )}-18 a K[1]^2\right )^{2/3}}{12 K[1] \sqrt [3]{\sqrt {3} \sqrt {K[1]^3 \left (27 a^2 K[1]-4 b^3\right )}-9 a K[1]^2}}dK[1]+c_1 \\
y(x)\to \int _1^x\frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) b K[2]+\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \left (2 \sqrt {3} \sqrt {K[2]^3 \left (27 a^2 K[2]-4 b^3\right )}-18 a K[2]^2\right )^{2/3}}{12 K[2] \sqrt [3]{\sqrt {3} \sqrt {K[2]^3 \left (27 a^2 K[2]-4 b^3\right )}-9 a K[2]^2}}dK[2]+c_1 \\
\end{align*}
✓ Sympy. Time used: 30.868 (sec). Leaf size: 393
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-a - b*Derivative(y(x), x) + x*Derivative(y(x), x)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} - \frac {\sqrt [3]{18} b \int \frac {1}{x \sqrt [3]{- \frac {9 a}{x} + \sqrt {3} \sqrt {\frac {27 a^{2} - \frac {4 b^{3}}{x}}{x^{2}}}}}\, dx}{3} - \frac {\sqrt [3]{12} \int \sqrt [3]{- \frac {9 a}{x} + \sqrt {3} \sqrt {\frac {27 a^{2} - \frac {4 b^{3}}{x}}{x^{2}}}}\, dx}{6}, \ y{\left (x \right )} = C_{1} + \frac {i \left (- 4 \sqrt [3]{2} \cdot 3^{\frac {2}{3}} b \int \frac {1}{x \sqrt [3]{- \frac {9 a}{x} + \sqrt {3} \sqrt {\frac {27 a^{2} - \frac {4 b^{3}}{x}}{x^{2}}}}}\, dx + \sqrt [3]{12} \int \sqrt [3]{- \frac {9 a}{x} + \sqrt {3} \sqrt {\frac {27 a^{2} - \frac {4 b^{3}}{x}}{x^{2}}}}\, dx - 2^{\frac {2}{3}} \cdot 3^{\frac {5}{6}} i \int \sqrt [3]{- \frac {9 a}{x} + \sqrt {3} \sqrt {\frac {27 a^{2} - \frac {4 b^{3}}{x}}{x^{2}}}}\, dx\right )}{6 \left (\sqrt {3} - i\right )}, \ y{\left (x \right )} = C_{1} - \frac {i \left (- 4 \sqrt [3]{2} \cdot 3^{\frac {2}{3}} b \int \frac {1}{x \sqrt [3]{- \frac {9 a}{x} + \sqrt {3} \sqrt {\frac {27 a^{2} - \frac {4 b^{3}}{x}}{x^{2}}}}}\, dx + \sqrt [3]{12} \int \sqrt [3]{- \frac {9 a}{x} + \sqrt {3} \sqrt {\frac {27 a^{2} - \frac {4 b^{3}}{x}}{x^{2}}}}\, dx + 2^{\frac {2}{3}} \cdot 3^{\frac {5}{6}} i \int \sqrt [3]{- \frac {9 a}{x} + \sqrt {3} \sqrt {\frac {27 a^{2} - \frac {4 b^{3}}{x}}{x^{2}}}}\, dx\right )}{6 \left (\sqrt {3} + i\right )}\right ]
\]