89.6.39 problem 40

Internal problem ID [24422]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Miscellaneous Exercises at page 45
Problem number : 40
Date solved : Thursday, October 02, 2025 at 10:27:41 PM
CAS classification : [_linear]

\begin{align*} y^{\prime }&=\cos \left (x \right )-y \sec \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 28

ode:=diff(y(x),x) = cos(x)-y(x)*sec(x); 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\left (-x +\cos \left (x \right )-2\right ) \left (-\cos \left (x \right )+\sin \left (x \right )-1\right )}{\cos \left (x \right )+\sin \left (x \right )+1} \]

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 24

ode=D[y[x],x]==Cos[x]-y[x]*Sec[x]; 
ic={y[0]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to (x-\cos (x)+2) e^{-2 \text {arctanh}\left (\tan \left (\frac {x}{2}\right )\right )} \end{align*}

✓ Sympy. Time used: 14.610 (sec). Leaf size: 180

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*sec(x) - cos(x) + Derivative(y(x), x),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\sqrt {\sin {\left (x \right )} - 1} \left (- \sqrt {\sin {\left (x \right )} - 1} \sqrt {\sin {\left (x \right )} + 1} - 2 \log {\left (2 \sqrt {\sin {\left (x \right )} - 1} + 2 \sqrt {\sin {\left (x \right )} + 1} \right )} + \int \limits ^{0} \frac {\sqrt {\sin {\left (x \right )} + 1} \sec {\left (x \right )}}{\sqrt {\sin {\left (x \right )} - 1}}\, dx - \int \left (- \frac {\sqrt {\sin {\left (x \right )} + 1} y{\left (x \right )} \sec {\left (x \right )}}{\sqrt {\sin {\left (x \right )} - 1}}\right )\, dx + \int \limits ^{0} \left (- \frac {\sqrt {\sin {\left (x \right )} + 1} y{\left (x \right )} \sec {\left (x \right )}}{\sqrt {\sin {\left (x \right )} - 1}}\right )\, dx + 3 \log {\left (2 \right )} + \frac {i \pi }{2} + 2 i\right )}{\sqrt {\sin {\left (x \right )} - 1} \int \frac {\sqrt {\sin {\left (x \right )} + 1} \sec {\left (x \right )}}{\sqrt {\sin {\left (x \right )} - 1}}\, dx - \sqrt {\sin {\left (x \right )} + 1}} \]