75.3.11 problem 11
Internal
problem
ID
[19915]
Book
:
A
short
course
on
differential
equations.
By
Donald
Francis
Campbell.
Maxmillan
company.
London.
1907
Section
:
Chapter
III.
Ordinary
differential
equations
of
the
first
order
and
first
degree.
Exercises
at
page
33
Problem
number
:
11
Date
solved
:
Thursday, October 02, 2025 at 05:01:05 PM
CAS
classification
:
[_Bernoulli]
\begin{align*} y^{\prime }+y \cos \left (x \right )&=y^{n} \sin \left (2 x \right ) \end{align*}
✓ Maple. Time used: 0.018 (sec). Leaf size: 49
ode:=diff(y(x),x)+cos(x)*y(x) = y(x)^n*sin(2*x);
dsolve(ode,y(x), singsol=all);
\[
y = \left (\frac {{\mathrm e}^{\sin \left (x \right ) \left (n -1\right )} c_1 n -{\mathrm e}^{\sin \left (x \right ) \left (n -1\right )} c_1 +2 \sin \left (x \right ) n -2 \sin \left (x \right )+2}{n -1}\right )^{-\frac {1}{n -1}}
\]
✓ Mathematica. Time used: 6.091 (sec). Leaf size: 36
ode=D[y[x],x]+Cos[x]*y[x]==y[x]^n*Sin[2*x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \left (c_1 e^{(n-1) \sin (x)}+\frac {2}{n-1}+2 \sin (x)\right ){}^{\frac {1}{1-n}} \end{align*}
✓ Sympy. Time used: 4.369 (sec). Leaf size: 228
from sympy import *
x = symbols("x")
n = symbols("n")
y = Function("y")
ode = Eq(y(x)*cos(x) - y(x)**n*sin(2*x) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \begin {cases} \left (\frac {C_{1} n^{2} e^{n \sin {\left (x \right )}}}{n^{2} e^{\sin {\left (x \right )}} - 2 n e^{\sin {\left (x \right )}} + e^{\sin {\left (x \right )}}} - \frac {2 C_{1} n e^{n \sin {\left (x \right )}}}{n^{2} e^{\sin {\left (x \right )}} - 2 n e^{\sin {\left (x \right )}} + e^{\sin {\left (x \right )}}} + \frac {C_{1} e^{n \sin {\left (x \right )}}}{n^{2} e^{\sin {\left (x \right )}} - 2 n e^{\sin {\left (x \right )}} + e^{\sin {\left (x \right )}}} + \frac {2 n^{2} \sin {\left (x \right )}}{n^{2} - 2 n + 1} - \frac {4 n \sin {\left (x \right )}}{n^{2} - 2 n + 1} + \frac {2 n}{n^{2} - 2 n + 1} + \frac {2 \sin {\left (x \right )}}{n^{2} - 2 n + 1} - \frac {2}{n^{2} - 2 n + 1}\right )^{- \frac {1}{n - 1}} & \text {for}\: n > 1 \vee n < 1 \\\left (C_{1} e^{n \sin {\left (x \right )} - \sin {\left (x \right )}} + n e^{n \sin {\left (x \right )} - \sin {\left (x \right )}} \cos ^{2}{\left (x \right )} - e^{n \sin {\left (x \right )} - \sin {\left (x \right )}} \cos ^{2}{\left (x \right )}\right )^{- \frac {1}{n - 1}} & \text {otherwise} \end {cases}
\]