78.6.12 problem 3 (b)
Internal
problem
ID
[18109]
Book
:
DIFFERENTIAL
EQUATIONS
WITH
APPLICATIONS
AND
HISTORICAL
NOTES
by
George
F.
Simmons.
3rd
edition.
2017.
CRC
press,
Boca
Raton
FL.
Section
:
Chapter
2.
First
order
equations.
Section
10
(Linear
equations).
Problems
at
page
82
Problem
number
:
3
(b)
Date
solved
:
Monday, March 31, 2025 at 05:10:39 PM
CAS
classification
:
[_Bernoulli]
\begin{align*} x y^{2} y^{\prime }+y^{3}&=x \cos \left (x \right ) \end{align*}
✓ Maple. Time used: 0.020 (sec). Leaf size: 110
ode:=x*y(x)^2*diff(y(x),x)+y(x)^3 = cos(x)*x;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {{\left (9 \left (x^{2}-2\right ) \cos \left (x \right )+3 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_1 \right )}^{{1}/{3}}}{x} \\
y &= -\frac {{\left (9 \left (x^{2}-2\right ) \cos \left (x \right )+3 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_1 \right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 x} \\
y &= \frac {{\left (9 \left (x^{2}-2\right ) \cos \left (x \right )+3 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_1 \right )}^{{1}/{3}} \left (-1+i \sqrt {3}\right )}{2 x} \\
\end{align*}
✓ Mathematica. Time used: 0.453 (sec). Leaf size: 114
ode=x*y[x]^2*D[y[x],x]+y[x]^3==x*Cos[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{3 x \left (x^2-6\right ) \sin (x)+9 \left (x^2-2\right ) \cos (x)+c_1}}{x} \\
y(x)\to -\frac {\sqrt [3]{-1} \sqrt [3]{3 x \left (x^2-6\right ) \sin (x)+9 \left (x^2-2\right ) \cos (x)+c_1}}{x} \\
y(x)\to \frac {(-1)^{2/3} \sqrt [3]{3 x \left (x^2-6\right ) \sin (x)+9 \left (x^2-2\right ) \cos (x)+c_1}}{x} \\
\end{align*}
✓ Sympy. Time used: 4.698 (sec). Leaf size: 139
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*y(x)**2*Derivative(y(x), x) - x*cos(x) + y(x)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{\frac {C_{1}}{x^{3}} + 3 \sin {\left (x \right )} + \frac {9 \cos {\left (x \right )}}{x} - \frac {18 \sin {\left (x \right )}}{x^{2}} - \frac {18 \cos {\left (x \right )}}{x^{3}}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{\frac {C_{1}}{x^{3}} + 3 \sin {\left (x \right )} + \frac {9 \cos {\left (x \right )}}{x} - \frac {18 \sin {\left (x \right )}}{x^{2}} - \frac {18 \cos {\left (x \right )}}{x^{3}}}}{2}, \ y{\left (x \right )} = \sqrt [3]{\frac {C_{1}}{x^{3}} + 3 \sin {\left (x \right )} + \frac {9 \cos {\left (x \right )}}{x} - \frac {18 \sin {\left (x \right )}}{x^{2}} - \frac {18 \cos {\left (x \right )}}{x^{3}}}\right ]
\]