68.12.40 problem 40

Internal problem ID [17634]
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 : 40
Date solved : Thursday, October 02, 2025 at 02:26:32 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=\frac {\csc \left (3 t \right )}{2} \end{align*}

With initial conditions

\begin{align*} y \left (\frac {\pi }{4}\right )&=\sqrt {2} \\ y^{\prime }\left (\frac {\pi }{4}\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.062 (sec). Leaf size: 38

ode:=diff(diff(y(t),t),t)+9*y(t) = 1/2*csc(3*t); 
ic:=[y(1/4*Pi) = 2^(1/2), D(y)(1/4*Pi) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = -\frac {\ln \left (\csc \left (3 t \right )\right ) \sin \left (3 t \right )}{18}+\frac {\left (-24-4 t +\pi \right ) \cos \left (3 t \right )}{24}+\frac {\sin \left (3 t \right ) \left (\ln \left (2\right )+36\right )}{36} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 40

ode=D[y[t],{t,2}]+9*y[t]==1/2*Csc[3*t]; 
ic={y[Pi/4]==Sqrt[2],Derivative[1][y][Pi/4]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{72} (3 (\pi -4 (t+6)) \cos (3 t)+2 \sin (3 t) (2 \log (\sin (3 t))+36+\log (2))) \end{align*}

✓ Sympy. Time used: 0.224 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(9*y(t) + Derivative(y(t), (t, 2)) - 1/(2*sin(3*t)),0) 
ics = {y(pi/4): sqrt(2), Subs(Derivative(y(t), t), t, pi/4): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {t}{6} - 1 + \frac {\pi }{24}\right ) \cos {\left (3 t \right )} + \left (\frac {\log {\left (\sin {\left (3 t \right )} \right )}}{18} + \frac {\log {\left (2 \right )}}{36} + 1\right ) \sin {\left (3 t \right )} \]