10.10.9 problem 9

Internal problem ID [1341]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 3, Second order linear equations, section 3.6, Variation of Parameters. page 190
Problem number : 9
Date solved : Saturday, March 29, 2025 at 10:52:42 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 33

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

\[ y = -4 \sin \left (t \right ) \ln \left (\sec \left (\frac {t}{2}\right )+\tan \left (\frac {t}{2}\right )\right )+\sin \left (t \right ) c_2 +\cos \left (t \right ) c_1 +8 \cos \left (\frac {t}{2}\right ) \]

✓ Mathematica. Time used: 0.084 (sec). Leaf size: 35

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

\[ y(t)\to -4 \sin (t) \text {arctanh}\left (\sin \left (\frac {t}{2}\right )\right )+8 \cos \left (\frac {t}{2}\right )+c_1 \cos (t)+c_2 \sin (t) \]

✓ Sympy. Time used: 0.821 (sec). Leaf size: 31

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

\[ y{\left (t \right )} = \left (C_{1} + 8 \cos {\left (\frac {t}{2} \right )}\right ) \cos {\left (t \right )} + \left (C_{2} + 2 \int \frac {\cos {\left (t \right )}}{\cos {\left (\frac {t}{2} \right )}}\, dt\right ) \sin {\left (t \right )} \]