68.12.6 problem 6

Internal problem ID [17600]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number : 6
Date solved : Thursday, October 02, 2025 at 02:26:06 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\frac {y}{4}&=\sec \left (\frac {t}{2}\right )+\csc \left (\frac {t}{2}\right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 48

ode:=diff(diff(y(t),t),t)+1/4*y(t) = sec(1/2*t)+csc(1/2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = 4 \cos \left (\frac {t}{2}\right ) \ln \left (\cos \left (\frac {t}{2}\right )\right )+4 \sin \left (\frac {t}{2}\right ) \ln \left (\sin \left (\frac {t}{2}\right )\right )+\left (-2 t +c_1 \right ) \cos \left (\frac {t}{2}\right )+2 \sin \left (\frac {t}{2}\right ) \left (t +\frac {c_2}{2}\right ) \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 50

ode=D[y[t],{t,2}]+1/4*y[t]==Sec[t/2]+Csc[t/2]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \cos \left (\frac {t}{2}\right ) \left (-2 t+4 \log \left (\cos \left (\frac {t}{2}\right )\right )+c_1\right )+\sin \left (\frac {t}{2}\right ) \left (2 t+4 \log \left (\sin \left (\frac {t}{2}\right )\right )+c_2\right ) \end{align*}

✓ Sympy. Time used: 0.259 (sec). Leaf size: 39

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

\[ y{\left (t \right )} = \left (C_{1} - 2 t + 4 \log {\left (\cos {\left (\frac {t}{2} \right )} \right )}\right ) \cos {\left (\frac {t}{2} \right )} + \left (C_{2} + 2 t + 4 \log {\left (\sin {\left (\frac {t}{2} \right )} \right )}\right ) \sin {\left (\frac {t}{2} \right )} \]