76.19.2 problem 2

Internal problem ID [17647]
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 : 2
Date solved : Thursday, March 13, 2025 at 10:45:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=t \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 9.806 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t)+2*y(t) = t; 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -\frac {5 \,{\mathrm e}^{-2 t}}{4}+\frac {t}{2}+3 \,{\mathrm e}^{-t}-\frac {3}{4} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 28

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

\[ y(t)\to \frac {1}{4} \left (2 t-5 e^{-2 t}+12 e^{-t}-3\right ) \]

✓ Sympy. Time used: 0.189 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t + 2*y(t) + 3*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {t}{2} - \frac {3}{4} + 3 e^{- t} - \frac {5 e^{- 2 t}}{4} \]