67.18.12 problem 27.1 (L)

Internal problem ID [16883]
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 (L)
Date solved : Thursday, October 02, 2025 at 01:40:06 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+13 y&=4 t +2 \,{\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: 0.225 (sec). Leaf size: 47

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

\[ y = -\frac {16}{169}+\frac {2 \cosh \left (2 t \right ) \left (346 \cos \left (3 t \right )+313 \sin \left (3 t \right )\right )}{169}+\frac {\left (-1423 \cos \left (3 t \right )-1226 \sin \left (3 t \right )\right ) \sinh \left (2 t \right )}{338}+\frac {4 t}{13} \]

✓ Mathematica. Time used: 0.529 (sec). Leaf size: 207

ode=D[y[t],{t,2}]+4*D[y[t],t]+13*y[t]==4*t+2*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 \cos (3 t) \int _1^0-\frac {2}{3} e^{2 K[2]} \sin (3 K[2]) \left (2 K[2]+e^{2 K[2]} \sin (3 K[2])\right )dK[2]+3 \cos (3 t) \int _1^t-\frac {2}{3} e^{2 K[2]} \sin (3 K[2]) \left (2 K[2]+e^{2 K[2]} \sin (3 K[2])\right )dK[2]-3 \sin (3 t) \int _1^0\frac {2}{3} e^{2 K[1]} \cos (3 K[1]) \left (2 K[1]+e^{2 K[1]} \sin (3 K[1])\right )dK[1]+3 \sin (3 t) \int _1^t\frac {2}{3} e^{2 K[1]} \cos (3 K[1]) \left (2 K[1]+e^{2 K[1]} \sin (3 K[1])\right )dK[1]+11 \sin (3 t)+12 \cos (3 t)\right ) \end{align*}

✓ Sympy. Time used: 0.281 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-4*t + 13*y(t) - 2*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)
 

\[ y{\left (t \right )} = \frac {4 t}{13} + \frac {\left (2 \sin {\left (3 t \right )} - 3 \cos {\left (3 t \right )}\right ) e^{2 t}}{52} + \left (\frac {1239 \sin {\left (3 t \right )}}{338} + \frac {2807 \cos {\left (3 t \right )}}{676}\right ) e^{- 2 t} - \frac {16}{169} \]