74.14.27 problem 27

Internal problem ID [16361]
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 : 27
Date solved : Monday, March 31, 2025 at 02:51:09 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime \prime }+y^{\prime \prime }&=\sec \left (t \right )^{2} \end{align*}

With initial conditions

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

✗ Maple

ode:=diff(diff(diff(diff(y(t),t),t),t),t)+diff(diff(y(t),t),t) = sec(t)^2; 
ic:=y(0) = 0, D(y)(0) = 1, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 60.05 (sec). Leaf size: 103

ode=D[y[t],{t,4}]+D[y[t],{t,2}]==Sec[t]^2; 
ic={y[0]==0,Derivative[1][y][0] ==1,Derivative[2][y][0] ==0,Derivative[3][y][0]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to t \left (-\int _1^0\left (\cos (K[1])+2 \text {arctanh}\left (\tan \left (\frac {K[1]}{2}\right )\right ) \sin (K[1])-1\right )dK[1]\right )+\int _1^t\int _1^{K[2]}\left (\cos (K[1])+2 \text {arctanh}\left (\tan \left (\frac {K[1]}{2}\right )\right ) \sin (K[1])-1\right )dK[1]dK[2]-\int _1^0\int _1^{K[2]}\left (\cos (K[1])+2 \text {arctanh}\left (\tan \left (\frac {K[1]}{2}\right )\right ) \sin (K[1])-1\right )dK[1]dK[2]+t \]

✓ Sympy. Time used: 1.523 (sec). Leaf size: 155

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 4)) - 1/cos(t)**2,0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1, Subs(Derivative(y(t), (t, 2)), t, 0): 0, Subs(Derivative(y(t), (t, 3)), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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