89.6.42 problem 43

Internal problem ID [24425]
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 : 43
Date solved : Thursday, October 02, 2025 at 10:27:45 PM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 18

ode:=diff(y(x),x) = tan(x)*y(x)+cos(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (2 c_1 +x \right ) \sec \left (x \right )}{2}+\frac {\sin \left (x \right )}{2} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 21

ode=D[y[x],x]==y[x]*Tan[x]+Cos[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} (\sin (x)+(x+2 c_1) \sec (x)) \end{align*}

✓ Sympy. Time used: 0.509 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = \frac {C_{1} + \frac {x}{2} + \frac {\sin {\left (2 x \right )}}{4}}{\cos {\left (x \right )}} \]