70.19.3 problem 3

Internal problem ID [19013]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.4 (Solving differential equations with Laplace transform). Problems at page 327
Problem number : 3
Date solved : Thursday, October 02, 2025 at 03:36:58 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-8 y^{\prime }+25 y&=0 \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.104 (sec). Leaf size: 13

ode:=diff(diff(y(t),t),t)-8*diff(y(t),t)+25*y(t) = 0; 
ic:=[y(0) = 0, D(y)(0) = 3]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = {\mathrm e}^{4 t} \sin \left (3 t \right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 15

ode=D[y[t],{t,2}]-8*D[y[t],t]+25*y[t]==0; 
ic={y[0]==0,Derivative[1][y][0] == 3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{4 t} \sin (3 t) \end{align*}

✓ Sympy. Time used: 0.109 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(25*y(t) - 8*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 3} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = e^{4 t} \sin {\left (3 t \right )} \]