29.3.19 problem 73
Internal
problem
ID
[4681]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
3
Problem
number
:
73
Date
solved
:
Sunday, March 30, 2025 at 03:36:25 AM
CAS
classification
:
[_linear]
\begin{align*} y^{\prime }&=y \sec \left (x \right )+\left (\sin \left (x \right )-1\right )^{2} \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 54
ode:=diff(y(x),x) = y(x)*sec(x)+(sin(x)-1)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \left (-\frac {11}{4}+4 \left (1+\sin \left (x \right )\right ) \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )+4 \left (1+\sin \left (x \right )\right ) \ln \left (\cos \left (x \right )\right )+\left (1+\sin \left (x \right )\right ) c_1 +\frac {\left (-2 \cos \left (x \right )^{2}-11\right ) \sin \left (x \right )}{4}+\frac {5 \cos \left (x \right )^{2}}{2}\right ) \sec \left (x \right )
\]
✓ Mathematica. Time used: 12.261 (sec). Leaf size: 55
ode=D[y[x],x]==y[x] Sec[x]+(Sin[x]-1)^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to -\frac {1}{4} e^{2 \text {arctanh}\left (\tan \left (\frac {x}{2}\right )\right )} (\cos (2 x)+4 (4 i x+3 \sin (x)-8 \log (\cos (x)+i (\sin (x)+1))-c_1))
\]
✓ Sympy. Time used: 141.595 (sec). Leaf size: 143
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-(sin(x) - 1)**2 - y(x)/cos(x) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \frac {\sqrt {\sin {\left (x \right )} + 1} \left (C_{1} + \int \frac {\sqrt {\sin {\left (x \right )} - 1}}{\sqrt {\sin {\left (x \right )} + 1}}\, dx - 2 \int \frac {\sqrt {\sin {\left (x \right )} - 1} \sin {\left (x \right )}}{\sqrt {\sin {\left (x \right )} + 1}}\, dx + \int \frac {\sqrt {\sin {\left (x \right )} - 1} \sin ^{2}{\left (x \right )}}{\sqrt {\sin {\left (x \right )} + 1}}\, dx + \int \frac {\sqrt {\sin {\left (x \right )} - 1} y{\left (x \right )}}{\sqrt {\sin {\left (x \right )} + 1} \cos {\left (x \right )}}\, dx\right )}{\sqrt {\sin {\left (x \right )} - 1} + \sqrt {\sin {\left (x \right )} + 1} \int \frac {\sqrt {\sin {\left (x \right )} - 1}}{\sqrt {\sin {\left (x \right )} + 1} \cos {\left (x \right )}}\, dx}
\]