82.4.14 problem 28-14

Internal problem ID [21832]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 28. Laplace transforms. Page 850
Problem number : 28-14
Date solved : Thursday, October 02, 2025 at 08:02:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-3 \\ y \left (1\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 15

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

\[ y = -2+t +{\mathrm e}^{-t} \left (t -1\right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 21

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

\begin{align*} y(t)&\to e^{-t} \left (e^t (t-2)+t-1\right ) \end{align*}

✓ Sympy. Time used: 0.045 (sec). Leaf size: 46

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

\[ y{\left (t \right )} = t + \frac {\left (-2 + 3 \cos {\left (\frac {\sqrt {3}}{3} \right )}\right ) \sin {\left (\frac {\sqrt {3} t}{3} \right )}}{\sin {\left (\frac {\sqrt {3}}{3} \right )}} - 3 \cos {\left (\frac {\sqrt {3} t}{3} \right )} \]