68.11.57 problem 70 (a)

Internal problem ID [17594]
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 : Thursday, October 02, 2025 at 02:26:01 PM
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*}

✓ Maple. Time used: 0.064 (sec). Leaf size: 54

ode:=4*diff(diff(y(t),t),t)+4*diff(y(t),t)+37*y(t) = cos(3*t); 
ic:=[y(0) = a, D(y)(Pi) = a]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 75

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

\begin{align*} 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 ) \end{align*}

✓ Sympy. Time used: 0.185 (sec). Leaf size: 61

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(37*y(t) - cos(3*t) + 4*Derivative(y(t), t) + 4*Derivative(y(t), (t, 2)),0) 
ics = {y(0): a, Subs(Derivative(y(t), t), t, pi): a} 
dsolve(ode,func=y(t),ics=ics)
 

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