74.12.45 problem 45

Internal problem ID [16352]
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 : Tuesday, January 28, 2025 at 09:06:32 AM
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*}

✓ Solution by Maple

Time used: 0.214 (sec). Leaf size: 46

dsolve([diff(y(t),t$2)+9*y(t)=csc(3*t),y(1/12*Pi) = 0, D(y)(1/12*Pi) = 1],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 51

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

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