67.18.11 problem 27.1 (k)

Internal problem ID [16882]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 27. Differentiation and the Laplace transform. Additional Exercises. page 496
Problem number : 27.1 (k)
Date solved : Thursday, October 02, 2025 at 01:40:05 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+13 y&={\mathrm e}^{2 t} \sin \left (3 t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=4 \\ y^{\prime }\left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.194 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)-4*diff(y(t),t)+13*y(t) = exp(2*t)*sin(3*t); 
ic:=[y(0) = 4, D(y)(0) = 3]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {{\mathrm e}^{2 t} \left (29 \sin \left (3 t \right )+3 \cos \left (3 t \right ) \left (-24+t \right )\right )}{18} \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 119

ode=D[y[t],{t,2}]-4*D[y[t],t]+13*y[t]==Exp[2*t]*Sin[3*t]; 
ic={y[0]==4,Derivative[1][y][0] ==3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{3} e^{2 t} \left (-3 \sin (3 t) \int _1^0\frac {1}{6} \sin (6 K[1])dK[1]+3 \sin (3 t) \int _1^t\frac {1}{6} \sin (6 K[1])dK[1]-3 \cos (3 t) \int _1^0-\frac {1}{3} \sin ^2(3 K[2])dK[2]+3 \cos (3 t) \int _1^t-\frac {1}{3} \sin ^2(3 K[2])dK[2]-5 \sin (3 t)+12 \cos (3 t)\right ) \end{align*}

✓ Sympy. Time used: 0.243 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(13*y(t) - exp(2*t)*sin(3*t) - 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 4, Subs(Derivative(y(t), t), t, 0): 3} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (4 - \frac {t}{6}\right ) \cos {\left (3 t \right )} - \frac {29 \sin {\left (3 t \right )}}{18}\right ) e^{2 t} \]