74.12.40 problem 40

Internal problem ID [16347]
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 : Tuesday, January 28, 2025 at 09:06:06 AM
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*}

✓ Solution by Maple

Time used: 0.191 (sec). Leaf size: 38

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

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

✓ Solution by Mathematica

Time used: 0.034 (sec). Leaf size: 40

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

\[ y(t)\to \frac {1}{72} (3 (\pi -4 (t+6)) \cos (3 t)+2 \sin (3 t) (2 \log (\sin (3 t))+36+\log (2))) \]