67.20.3 problem 29.6 (c)

Internal problem ID [16899]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 29. Convolution. Additional Exercises. page 523
Problem number : 29.6 (c)
Date solved : Thursday, October 02, 2025 at 01:40:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y&={\mathrm e}^{3 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: 0.130 (sec). Leaf size: 23

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

\[ y = \frac {{\mathrm e}^{3 t}}{13}-\frac {\cos \left (2 t \right )}{13}-\frac {3 \sin \left (2 t \right )}{26} \]

✓ Mathematica. Time used: 0.067 (sec). Leaf size: 111

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

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

✓ Sympy. Time used: 0.056 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - exp(3*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)
 

\[ y{\left (t \right )} = \frac {e^{3 t}}{13} - \frac {3 \sin {\left (2 t \right )}}{26} - \frac {\cos {\left (2 t \right )}}{13} \]