31.1.22 problem 10.4
Internal
problem
ID
[5720]
Book
:
Differential
Equations,
By
George
Boole
F.R.S.
1865
Section
:
Chapter
2
Problem
number
:
10.4
Date
solved
:
Sunday, March 30, 2025 at 10:06:02 AM
CAS
classification
:
[_Bernoulli]
\begin{align*} z^{\prime }+z \cos \left (x \right )&=z^{n} \sin \left (2 x \right ) \end{align*}
✓ Maple. Time used: 0.086 (sec). Leaf size: 49
ode:=diff(z(x),x)+z(x)*cos(x) = z(x)^n*sin(2*x);
dsolve(ode,z(x), singsol=all);
\[
z = \left (\frac {{\mathrm e}^{\sin \left (x \right ) \left (n -1\right )} c_1 n -{\mathrm e}^{\sin \left (x \right ) \left (n -1\right )} c_1 +2 \sin \left (x \right ) n -2 \sin \left (x \right )+2}{n -1}\right )^{-\frac {1}{n -1}}
\]
✓ Mathematica. Time used: 6.724 (sec). Leaf size: 36
ode=D[z[x],x]+z[x]*Cos[x]==z[x]^n*Sin[2*x];
ic={};
DSolve[{ode,ic},z[x],x,IncludeSingularSolutions->True]
\[
z(x)\to \left (c_1 e^{(n-1) \sin (x)}+\frac {2}{n-1}+2 \sin (x)\right ){}^{\frac {1}{1-n}}
\]
✓ Sympy. Time used: 6.727 (sec). Leaf size: 228
from sympy import *
x = symbols("x")
n = symbols("n")
z = Function("z")
ode = Eq(z(x)*cos(x) - z(x)**n*sin(2*x) + Derivative(z(x), x),0)
ics = {}
dsolve(ode,func=z(x),ics=ics)
\[
z{\left (x \right )} = \begin {cases} \left (\frac {C_{1} n^{2} e^{n \sin {\left (x \right )}}}{n^{2} e^{\sin {\left (x \right )}} - 2 n e^{\sin {\left (x \right )}} + e^{\sin {\left (x \right )}}} - \frac {2 C_{1} n e^{n \sin {\left (x \right )}}}{n^{2} e^{\sin {\left (x \right )}} - 2 n e^{\sin {\left (x \right )}} + e^{\sin {\left (x \right )}}} + \frac {C_{1} e^{n \sin {\left (x \right )}}}{n^{2} e^{\sin {\left (x \right )}} - 2 n e^{\sin {\left (x \right )}} + e^{\sin {\left (x \right )}}} + \frac {2 n^{2} \sin {\left (x \right )}}{n^{2} - 2 n + 1} - \frac {4 n \sin {\left (x \right )}}{n^{2} - 2 n + 1} + \frac {2 n}{n^{2} - 2 n + 1} + \frac {2 \sin {\left (x \right )}}{n^{2} - 2 n + 1} - \frac {2}{n^{2} - 2 n + 1}\right )^{- \frac {1}{n - 1}} & \text {for}\: n > 1 \vee n < 1 \\\left (C_{1} e^{n \sin {\left (x \right )} - \sin {\left (x \right )}} - n e^{n \sin {\left (x \right )} - \sin {\left (x \right )}} \sin ^{2}{\left (x \right )} + e^{n \sin {\left (x \right )} - \sin {\left (x \right )}} \sin ^{2}{\left (x \right )}\right )^{- \frac {1}{n - 1}} & \text {otherwise} \end {cases}
\]