50.3.18 problem 3(b)
Internal
problem
ID
[7842]
Book
:
Differential
Equations:
Theory,
Technique,
and
Practice
by
George
Simmons,
Steven
Krantz.
McGraw-Hill
NY.
2007.
1st
Edition.
Section
:
Chapter
1.
What
is
a
differential
equation.
Section
1.4
First
Order
Linear
Equations.
Page
15
Problem
number
:
3(b)
Date
solved
:
Wednesday, March 05, 2025 at 05:07:36 AM
CAS
classification
:
[_Bernoulli]
\begin{align*} x y^{2} y^{\prime }+y^{3}&=x \cos \left (x \right ) \end{align*}
✓ Maple. Time used: 0.040 (sec). Leaf size: 110
ode:=x*y(x)^2*diff(y(x),x)+y(x)^3 = x*cos(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 (i \sqrt {3}-1\right )}{2 x} \\
\end{align*}
✓ Mathematica. Time used: 0.52 (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: 5.144 (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)
\[
\left [ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{\frac {C_{1}}{x^{3}} + 3 \sin {\left (x \right )} + \frac {9 \cos {\left (x \right )}}{x} - \frac {18 \sin {\left (x \right )}}{x^{2}} - \frac {18 \cos {\left (x \right )}}{x^{3}}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{\frac {C_{1}}{x^{3}} + 3 \sin {\left (x \right )} + \frac {9 \cos {\left (x \right )}}{x} - \frac {18 \sin {\left (x \right )}}{x^{2}} - \frac {18 \cos {\left (x \right )}}{x^{3}}}}{2}, \ y{\left (x \right )} = \sqrt [3]{\frac {C_{1}}{x^{3}} + 3 \sin {\left (x \right )} + \frac {9 \cos {\left (x \right )}}{x} - \frac {18 \sin {\left (x \right )}}{x^{2}} - \frac {18 \cos {\left (x \right )}}{x^{3}}}\right ]
\]