76.17.8 problem 17

Internal problem ID [17623]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.7 (Variation of parameters). Problems at page 280
Problem number : 17
Date solved : Monday, March 31, 2025 at 04:22:51 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 68

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

\[ y = 8 \ln \left (\csc \left (\frac {t}{2}\right )-\cot \left (\frac {t}{2}\right )\right ) \left (2 \cos \left (\frac {t}{2}\right )^{3}-\cos \left (\frac {t}{2}\right )\right ) \sin \left (\frac {t}{2}\right )+16 \cos \left (\frac {t}{2}\right )^{2} \sin \left (\frac {t}{2}\right )+\cos \left (2 t \right ) c_1 +\sin \left (2 t \right ) c_2 -\frac {8 \sin \left (\frac {t}{2}\right )}{3} \]

✓ Mathematica. Time used: 0.103 (sec). Leaf size: 64

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

\[ y(t)\to \frac {4}{3} \sin \left (\frac {t}{2}\right )+4 \sin \left (\frac {3 t}{2}\right )+2 \sin (2 t) \log \left (\sin \left (\frac {t}{4}\right )\right )+c_1 \cos (2 t)+c_2 \sin (2 t)-2 \sin (2 t) \log \left (\cos \left (\frac {t}{4}\right )\right ) \]

✓ Sympy. Time used: 3.476 (sec). Leaf size: 39

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

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