74.14.28 problem 28

Internal problem ID [16362]
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 : 28
Date solved : Monday, March 31, 2025 at 02:51:10 PM
CAS classification : [[_3rd_order, _missing_y]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.206 (sec). Leaf size: 48

ode:=diff(diff(diff(y(t),t),t),t)+diff(y(t),t) = sec(t); 
ic:=y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 1; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \frac {\left (-i \ln \left (\sec \left (t \right )\right )-t -1\right ) {\mathrm e}^{-i t}}{2}-2 i \arctan \left ({\mathrm e}^{i t}\right )+\frac {\left (i \ln \left (\sec \left (t \right )\right )-t -1\right ) {\mathrm e}^{i t}}{2}+\frac {i \pi }{2}+1 \]

✓ Mathematica. Time used: 60.029 (sec). Leaf size: 56

ode=D[ y[t],{t,3}]+D[y[t],t]==Sec[t]; 
ic={y[0]==0,Derivative[1][y][0] ==0,Derivative[2][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \int _1^t(\cos (K[1]) \log (\cos (K[1]))+(K[1]+1) \sin (K[1]))dK[1]-\int _1^0(\cos (K[1]) \log (\cos (K[1]))+(K[1]+1) \sin (K[1]))dK[1] \]

✓ Sympy. Time used: 0.405 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(Derivative(y(t), t) + Derivative(y(t), (t, 3)) - 1/cos(t),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0, Subs(Derivative(y(t), (t, 2)), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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