61.3.13 problem Problem 14

Internal problem ID [15312]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.5 Laplace transform. Homogeneous equations. Problems page 357
Problem number : Problem 14
Date solved : Thursday, October 02, 2025 at 10:11:32 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+5 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.083 (sec). Leaf size: 12

ode:=diff(diff(y(t),t),t)-2*diff(y(t),t)+5*y(t) = 0; 
ic:=[y(0) = 0, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 14

ode=D[y[t],{t,2}]-2*D[y[t],t]+5*y[t]==0; 
ic={y[0]==0,Derivative[1][y][0] ==-1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.100 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*y(t) - 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

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