74.14.15 problem 15

Internal problem ID [16341]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.6, page 187
Problem number : 15
Date solved : Thursday, March 13, 2025 at 08:10:46 AM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }+9 y^{\prime }&=\sec \left (3 t \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 115

ode:=diff(diff(diff(y(t),t),t),t)+9*diff(y(t),t) = sec(3*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {i \left ({\mathrm e}^{3 i t}-{\mathrm e}^{-3 i t}\right ) \ln \left (\frac {{\mathrm e}^{3 i t}}{{\mathrm e}^{6 i t}+1}\right )}{54}-\frac {i \arctan \left (2 \,{\mathrm e}^{i t}-\sqrt {3}\right )}{27}-\frac {i \arctan \left (2 \,{\mathrm e}^{i t}+\sqrt {3}\right )}{27}-\frac {i \arctan \left ({\mathrm e}^{3 i t}\right )}{27}-\frac {i {\mathrm e}^{-3 i t}}{54}+\frac {i {\mathrm e}^{3 i t}}{54}+\frac {i \arctan \left ({\mathrm e}^{i t}\right )}{27}+\frac {\left (1+9 c_{1} -\ln \left (2\right )\right ) \sin \left (3 t \right )}{27}+\frac {\left (-t -3 c_{2} \right ) \cos \left (3 t \right )}{9}+c_{3} \]

✓ Mathematica. Time used: 60.048 (sec). Leaf size: 51

ode=D[ y[t],{t,3}]+9*D[y[t],t]==Sec[3*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \int _1^t\left (\cos (3 K[1]) \left (c_1+\frac {1}{9} \log (\cos (3 K[1]))\right )+\frac {1}{3} (3 c_2+K[1]) \sin (3 K[1])\right )dK[1]+c_3 \]

✓ Sympy. Time used: 0.497 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(9*Derivative(y(t), t) + Derivative(y(t), (t, 3)) - 1/cos(3*t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} + \left (C_{2} - \frac {t}{9}\right ) \cos {\left (3 t \right )} + \left (C_{3} + \frac {\log {\left (\cos {\left (3 t \right )} \right )}}{27}\right ) \sin {\left (3 t \right )} - \frac {\log {\left (\sin {\left (3 t \right )} - 1 \right )}}{54} + \frac {\log {\left (\sin {\left (3 t \right )} + 1 \right )}}{54} \]