89.33.14 problem 15

Internal problem ID [24998]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 17. Special Equations of order Two. Exercises at page 251
Problem number : 15
Date solved : Thursday, October 02, 2025 at 11:46:08 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 20

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

\[ y = c_1 +\left (\ln \left (\sec \left (x \right )+\tan \left (x \right )\right )-\ln \left (\cos \left (x \right )\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.1 (sec). Leaf size: 25

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

\begin{align*} y(x)&\to c_1 \log \left (e^{4 \text {arctanh}\left (\tan \left (\frac {x}{2}\right )\right )}+1\right )+c_2 \end{align*}

✓ Sympy. Time used: 0.992 (sec). Leaf size: 24

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

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