90.10.2 problem 18

Internal problem ID [25196]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 2. The Laplace Transform. Exercises at page 149
Problem number : 18
Date solved : Thursday, October 02, 2025 at 11:57:43 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+9 y&=50 \sin \left (t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.047 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+6*diff(y(t),t)+9*y(t) = 50*sin(t); 
ic:=[y(0) = 0, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -3 \cos \left (t \right )+4 \sin \left (t \right )+{\mathrm e}^{-3 t} \left (3+7 t \right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 30

ode=D[y[t],{t,2}]+6*D[y[t],{t,1}]+9*y[t]==50*Sin[t]; 
ic={y[0]==0,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-3 t} \left (7 t+4 e^{3 t} \sin (t)+3\right )-3 \cos (t) \end{align*}

✓ Sympy. Time used: 0.054 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(9*y(t) - 50*sin(t) + 7*Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 25 \sin {\left (t \right )} - \frac {23 \sqrt {7} \sin {\left (\frac {3 \sqrt {7} t}{7} \right )}}{3} \]