89.5.10 problem 10

Internal problem ID [24360]
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. Exercises at page 43
Problem number : 10
Date solved : Thursday, October 02, 2025 at 10:22:06 PM
CAS classification : [_linear]

\begin{align*} y-\cos \left (x \right )^{2}+\cos \left (x \right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 28

ode:=y(x)-cos(x)^2+cos(x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (x -\cos \left (x \right )+c_1 \right ) \left (\cos \left (x \right )-\sin \left (x \right )+1\right )}{\cos \left (x \right )+\sin \left (x \right )+1} \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 25

ode=(y[x]-Cos[x]^2)+( Cos[x] )*D[y[x],x]== 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-2 \text {arctanh}\left (\tan \left (\frac {x}{2}\right )\right )} (x-\cos (x)+c_1) \end{align*}

✓ Sympy. Time used: 16.878 (sec). Leaf size: 138

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) - cos(x)**2 + cos(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1} \sqrt {\sin {\left (x \right )} - 1} - 2 \sqrt {\sin {\left (x \right )} - 1} \log {\left (2 \sqrt {\sin {\left (x \right )} - 1} + 2 \sqrt {\sin {\left (x \right )} + 1} \right )} + \sqrt {\sin {\left (x \right )} - 1} \int \frac {\sqrt {\sin {\left (x \right )} + 1} y{\left (x \right )}}{\sqrt {\sin {\left (x \right )} - 1} \cos {\left (x \right )}}\, dx - \sqrt {\sin {\left (x \right )} + 1} \sin {\left (x \right )} + \sqrt {\sin {\left (x \right )} + 1}}{\sqrt {\sin {\left (x \right )} - 1} \int \frac {\sqrt {\sin {\left (x \right )} + 1}}{\sqrt {\sin {\left (x \right )} - 1} \cos {\left (x \right )}}\, dx - \sqrt {\sin {\left (x \right )} + 1}} \]