68.12.45 problem 45

Internal problem ID [17639]
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 : 45
Date solved : Thursday, October 02, 2025 at 02:26:39 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=\csc \left (3 t \right ) \end{align*}

With initial conditions

\begin{align*} y \left (\frac {\pi }{12}\right )&=0 \\ y^{\prime }\left (\frac {\pi }{12}\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.117 (sec). Leaf size: 46

ode:=diff(diff(y(t),t),t)+9*y(t) = csc(3*t); 
ic:=[y(1/12*Pi) = 0, D(y)(1/12*Pi) = 1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = -\frac {\ln \left (\csc \left (3 t \right )\right ) \sin \left (3 t \right )}{9}+\frac {\left (-12 t +\pi -6 \sqrt {2}\right ) \cos \left (3 t \right )}{36}+\frac {\sin \left (3 t \right ) \left (\ln \left (2\right )+3 \sqrt {2}\right )}{18} \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 51

ode=D[y[t],{t,2}]+9*y[t]==Csc[3*t]; 
ic={y[Pi/12]==0,Derivative[1][y][Pi/12]==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{36} \left (\left (\pi -6 \left (2 t+\sqrt {2}\right )\right ) \cos (3 t)+2 \sin (3 t) \left (2 \log (\sin (3 t))+3 \sqrt {2}+\log (2)\right )\right ) \end{align*}

✓ Sympy. Time used: 0.217 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(9*y(t) + Derivative(y(t), (t, 2)) - 1/sin(3*t),0) 
ics = {y(pi/12): 0, Subs(Derivative(y(t), t), t, pi/12): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {t}{3} - \frac {\sqrt {2}}{6} + \frac {\pi }{36}\right ) \cos {\left (3 t \right )} + \left (\frac {\log {\left (\sin {\left (3 t \right )} \right )}}{9} + \frac {\log {\left (2 \right )}}{18} + \frac {\sqrt {2}}{6}\right ) \sin {\left (3 t \right )} \]