76.22.5 problem 18

Internal problem ID [17707]
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 5.8 (Convolution Integrals and Their Applications). Problems at page 359
Problem number : 18
Date solved : Thursday, March 13, 2025 at 10:47:47 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=g \left (t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=-3 \end{align*}

✓ Maple. Time used: 14.378 (sec). Leaf size: 37

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+4*y(t) = g(t); 
ic:=y(0) = 2, D(y)(0) = -3; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -\int _{0}^{t}\left (-t +\textit {\_U1} \right ) {\mathrm e}^{-2 t +2 \textit {\_U1}} g \left (\textit {\_U1} \right )d \textit {\_U1} +\left (t +2\right ) {\mathrm e}^{-2 t} \]

✓ Mathematica. Time used: 0.066 (sec). Leaf size: 95

ode=D[y[t],{t,2}]+4*D[y[t],t]+4*y[t]==g[t]; 
ic={y[0]==2,Derivative[1][y][0] ==-3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{-2 t} \left (t \left (-\int _1^0e^{2 K[2]} g(K[2])dK[2]\right )+t \int _1^te^{2 K[2]} g(K[2])dK[2]+\int _1^t-e^{2 K[1]} g(K[1]) K[1]dK[1]-\int _1^0-e^{2 K[1]} g(K[1]) K[1]dK[1]+t+2\right ) \]

✓ Sympy. Time used: 1.337 (sec). Leaf size: 56

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-g(t) + 4*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): -3} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (t \left (\int g{\left (t \right )} e^{2 t}\, dt - \int \limits ^{0} g{\left (t \right )} e^{2 t}\, dt + 1\right ) - \int t g{\left (t \right )} e^{2 t}\, dt + \int \limits ^{0} t g{\left (t \right )} e^{2 t}\, dt + 2\right ) e^{- 2 t} \]