14.21.27 problem Problem 27

Internal problem ID [3954]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.4. page 689
Problem number : Problem 27
Date solved : Tuesday, September 30, 2025 at 06:59:24 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=7 \sin \left (4 t \right )+14 \cos \left (4 t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.131 (sec). Leaf size: 29

ode:=diff(diff(y(t),t),t)+9*y(t) = 7*sin(4*t)+14*cos(4*t); 
ic:=[y(0) = 1, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -2 \cos \left (4 t \right )-\sin \left (4 t \right )+3 \cos \left (3 t \right )+2 \sin \left (3 t \right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 49

ode=D[y[t],{t,2}]+8*y[t]==7*Sin[4*t]+14*Cos[4*t]; 
ic={y[0]==1,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{8} \left (-7 \sin (4 t)+11 \sqrt {2} \sin \left (2 \sqrt {2} t\right )-14 \cos (4 t)+22 \cos \left (2 \sqrt {2} t\right )\right ) \end{align*}

✓ Sympy. Time used: 0.056 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(9*y(t) - 7*sin(4*t) - 14*cos(4*t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 2 \sin {\left (3 t \right )} - \sin {\left (4 t \right )} + 3 \cos {\left (3 t \right )} - 2 \cos {\left (4 t \right )} \]