74.12.31 problem 31

Internal problem ID [16267]
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 : 31
Date solved : Monday, March 31, 2025 at 02:48:53 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 55

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

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

✓ Mathematica. Time used: 0.342 (sec). Leaf size: 73

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

\[ y(t)\to \cos (t) \int _1^t-2 \left (\cos \left (\frac {K[1]}{2}\right )+\sin \left (\frac {K[1]}{2}\right )\right )dK[1]+\sin (t) \int _1^t2 \cot (K[2]) \left (\cos \left (\frac {K[2]}{2}\right )+\sin \left (\frac {K[2]}{2}\right )\right )dK[2]+c_1 \cos (t)+c_2 \sin (t) \]

✓ Sympy. Time used: 1.209 (sec). Leaf size: 48

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

\[ y{\left (t \right )} = \left (C_{1} + 4 \sqrt {2} \cos {\left (\frac {t}{2} + \frac {\pi }{4} \right )}\right ) \cos {\left (t \right )} + \left (C_{2} + \int \frac {\cos {\left (t \right )}}{\sin {\left (\frac {t}{2} \right )}}\, dt + \int \frac {\cos {\left (t \right )}}{\cos {\left (\frac {t}{2} \right )}}\, dt\right ) \sin {\left (t \right )} \]