81.12.14 problem 16-15

Internal problem ID [21667]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 16. Variation of Parameters. Page 375.
Problem number : 16-15
Date solved : Thursday, October 02, 2025 at 07:59:38 PM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 63

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

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

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 57

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

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

✓ Sympy. Time used: 0.201 (sec). Leaf size: 37

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

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