26.6.37 problem Exercise 12.37, page 103

Internal problem ID [7037]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous Methods
Problem number : Exercise 12.37, page 103
Date solved : Tuesday, September 30, 2025 at 04:16:48 PM
CAS classification : [_linear]

\begin{align*} y^{\prime } \cos \left (x \right )+y+\left (1+\sin \left (x \right )\right ) \cos \left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 29

ode:=cos(x)*diff(y(x),x)+y(x)+(sin(x)+1)*cos(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.767 (sec). Leaf size: 46

ode=D[y[x],x]*Cos[x]+y[x]+(1+Sin[x])*Cos[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 )} \left (\int _1^x-e^{2 \text {arctanh}\left (\tan \left (\frac {K[1]}{2}\right )\right )} (\sin (K[1])+1)dK[1]+c_1\right ) \end{align*}

✓ Sympy. Time used: 19.832 (sec). Leaf size: 87

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

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