89.26.8 problem 10

Internal problem ID [24875]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 11. Variation of parameters and other methods. Miscellaneous Exercises at page 177
Problem number : 10
Date solved : Thursday, October 02, 2025 at 10:48:57 PM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 89

ode:=diff(diff(diff(y(x),x),x),x)+diff(y(x),x) = sec(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-{\mathrm e}^{-i x}-{\mathrm e}^{i x}\right ) \ln \left (\frac {i {\mathrm e}^{i x}-1}{-{\mathrm e}^{i x}+i}\right )}{2}+\frac {\left (i c_1 -c_2 \right ) {\mathrm e}^{-i x}}{2}-i \ln \left ({\mathrm e}^{i x}\right )+\frac {\left (-i c_1 -c_2 \right ) {\mathrm e}^{i x}}{2}-x +c_3 \]

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 30

ode=D[y[x],{x,3}]+D[y[x],{x,1}]== Sec[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -2 \cos (x) \text {arctanh}\left (\tan \left (\frac {x}{2}\right )\right )-c_2 \cos (x)+c_1 \sin (x)+c_3 \end{align*}

✓ Sympy. Time used: 0.223 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sec(x)**2 + Derivative(y(x), x) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + C_{3} \sin {\left (x \right )} + \left (C_{2} + \frac {\log {\left (\sin {\left (x \right )} - 1 \right )}}{2} - \frac {\log {\left (\sin {\left (x \right )} + 1 \right )}}{2}\right ) \cos {\left (x \right )} \]