46.5.8 problem 38

Internal problem ID [9617]
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. 7.2.2 TRANSFORMS OF DERIVATIVES Page 289
Problem number : 38
Date solved : Tuesday, September 30, 2025 at 06:21:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+9 y&={\mathrm e}^{t} \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.122 (sec). Leaf size: 21

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

\[ y = \frac {{\mathrm e}^{t}}{10}-\frac {\cos \left (3 t \right )}{10}-\frac {\sin \left (3 t \right )}{30} \]

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 105

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

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

✓ Sympy. Time used: 0.048 (sec). Leaf size: 20

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

\[ y{\left (t \right )} = \frac {e^{t}}{10} - \frac {\sin {\left (3 t \right )}}{30} - \frac {\cos {\left (3 t \right )}}{10} \]