74.11.57 problem 70 (a)

Internal problem ID [16307]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.3, page 156
Problem number : 70 (a)
Date solved : Tuesday, January 28, 2025 at 09:02:21 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y^{\prime \prime }+4 y^{\prime }+37 y&=\cos \left (3 t \right ) \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.155 (sec). Leaf size: 55

dsolve([4*diff(y(t),t$2)+4*diff(y(t),t)+37*y(t)=cos(3*t),y(0) = a, D(y)(Pi) = a],y(t), singsol=all)
 

\[ y = \frac {\left (-290 a -72\right ) \sin \left (3 t \right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{2}}}{870}+\left (a -\frac {1}{145}\right ) \left (\frac {\sin \left (3 t \right )}{6}+\cos \left (3 t \right )\right ) {\mathrm e}^{-\frac {t}{2}}+\frac {\cos \left (3 t \right )}{145}+\frac {12 \sin \left (3 t \right )}{145} \]

✓ Solution by Mathematica

Time used: 0.027 (sec). Leaf size: 75

DSolve[{4*D[y[t],{t,2}]+4*D[y[t],t]+37*y[t]==Cos[3*t],{y[0]==a,Derivative[1][y][Pi]==a}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{870} e^{-t/2} \left (6 \left (145 a+e^{t/2}-1\right ) \cos (3 t)-\left (145 \left (2 e^{\pi /2}-1\right ) a-72 e^{t/2}+72 e^{\pi /2}+1\right ) \sin (3 t)\right ) \]