46.7.3 problem 11

Internal problem ID [9644]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. 7.4.1 DERIVATIVES OF A TRANSFORM. Page 309
Problem number : 11
Date solved : Tuesday, September 30, 2025 at 06:21:48 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=\cos \left (3 t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=5 \\ \end{align*}

✓ Maple. Time used: 0.170 (sec). Leaf size: 20

ode:=diff(diff(y(t),t),t)+9*y(t) = cos(3*t); 
ic:=[y(0) = 2, D(y)(0) = 5]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 2 \cos \left (3 t \right )+\frac {\sin \left (3 t \right ) \left (10+t \right )}{6} \]

✓ Mathematica. Time used: 0.046 (sec). Leaf size: 110

ode=D[y[t],{t,2}]+9*y[t]==Cos[3*t]; 
ic={y[0]==2,Derivative[1][y][0] ==5}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\sin (3 t) \int _1^0\frac {1}{3} \cos ^2(3 K[2])dK[2]+\sin (3 t) \int _1^t\frac {1}{3} \cos ^2(3 K[2])dK[2]+\cos (3 t) \left (-\int _1^0-\frac {1}{6} \sin (6 K[1])dK[1]\right )+\cos (3 t) \int _1^t-\frac {1}{6} \sin (6 K[1])dK[1]+\frac {5}{3} \sin (3 t)+2 \cos (3 t) \end{align*}

✓ Sympy. Time used: 0.062 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(9*y(t) - cos(3*t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): 5} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {t}{6} + \frac {5}{3}\right ) \sin {\left (3 t \right )} + 2 \cos {\left (3 t \right )} \]