76.19.9 problem 9

Internal problem ID [17654]
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 : 9
Date solved : Thursday, March 13, 2025 at 10:46:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+2 y&={\mathrm e}^{-t} \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: 12.321 (sec). Leaf size: 24

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

\[ y = \frac {{\mathrm e}^{-t}}{5}+\frac {\left (-\cos \left (t \right )+7 \sin \left (t \right )\right ) {\mathrm e}^{t}}{5} \]

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 29

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

\[ y(t)\to \frac {1}{5} \left (e^{-t}+7 e^t \sin (t)-e^t \cos (t)\right ) \]

✓ Sympy. Time used: 0.240 (sec). Leaf size: 24

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

\[ y{\left (t \right )} = \left (\frac {7 \sin {\left (t \right )}}{5} - \frac {\cos {\left (t \right )}}{5}\right ) e^{t} + \frac {e^{- t}}{5} \]