8.21.9 problem 9

Internal problem ID [2718]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.15, Higher order equations. Excercises page 263
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 05:50:11 AM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 86

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

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

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 35

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

\begin{align*} y(t)&\to -\sin (t) \text {arctanh}(\sin (t))-\frac {1}{2} \log \left (\cos ^2(t)\right )-c_2 \cos (t)+c_1 \sin (t)+c_3 \end{align*}

✓ Sympy. Time used: 0.220 (sec). Leaf size: 36

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

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