67.16.20 problem 24.4 (b)

Internal problem ID [16819]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 24. Variation of parameters. Additional exercises page 444
Problem number : 24.4 (b)
Date solved : Thursday, October 02, 2025 at 01:39:23 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 46

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

\[ y = \frac {1}{2}+\frac {\int {\mathrm e}^{-x} \tan \left (x \right )d x {\mathrm e}^{x}}{2}+\frac {\left (\cos \left (x \right )-\sin \left (x \right )\right ) \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )}{2}+c_2 \,{\mathrm e}^{x}+c_1 \cos \left (x \right )+c_3 \sin \left (x \right ) \]

✓ Mathematica. Time used: 0.188 (sec). Leaf size: 97

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

\begin{align*} y(x)&\to e^x \int _1^x\frac {1}{2} e^{-K[3]} \tan (K[3])dK[3]+\cos (x) \int _1^x\frac {1}{2} (\sin (K[1]) \tan (K[1])-\sin (K[1]))dK[1]+\sin (x) \int _1^x-\frac {1}{2} (\cos (K[2])+\sin (K[2])) \tan (K[2])dK[2]+c_3 e^x+c_1 \cos (x)+c_2 \sin (x) \end{align*}

✓ Sympy. Time used: 0.853 (sec). Leaf size: 65

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

\[ y{\left (x \right )} = \left (C_{1} + \frac {\int e^{- x} \tan {\left (x \right )}\, dx}{2}\right ) e^{x} + \left (C_{2} - \frac {\log {\left (\sin {\left (x \right )} - 1 \right )}}{4} + \frac {\log {\left (\sin {\left (x \right )} + 1 \right )}}{4}\right ) \cos {\left (x \right )} + \left (C_{3} + \frac {\log {\left (\sin {\left (x \right )} - 1 \right )}}{4} - \frac {\log {\left (\sin {\left (x \right )} + 1 \right )}}{4}\right ) \sin {\left (x \right )} + \frac {1}{2} \]