11.5.12 problem 19(a)

Internal problem ID [1517]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.5, The Laplace Transform. Impulse functions. page 273
Problem number : 19(a)
Date solved : Saturday, March 29, 2025 at 10:57:22 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+2 y&=f \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.044 (sec). Leaf size: 43

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+2*y(t) = f(t); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = {\mathrm e}^{-t} \left (\int _{0}^{t}{\mathrm e}^{\textit {\_z1}} \cos \left (\textit {\_z1} \right ) f \left (\textit {\_z1} \right )d \textit {\_z1} \sin \left (t \right )-\int _{0}^{t}{\mathrm e}^{\textit {\_z1}} \sin \left (\textit {\_z1} \right ) f \left (\textit {\_z1} \right )d \textit {\_z1} \cos \left (t \right )\right ) \]

✓ Mathematica. Time used: 0.106 (sec). Leaf size: 99

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

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

✓ Sympy. Time used: 3.428 (sec). Leaf size: 60

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

\[ y{\left (t \right )} = \left (\left (- \int f{\left (t \right )} e^{t} \sin {\left (t \right )}\, dt + \int \limits ^{0} f{\left (t \right )} e^{t} \sin {\left (t \right )}\, dt\right ) \cos {\left (t \right )} + \left (\int f{\left (t \right )} e^{t} \cos {\left (t \right )}\, dt - \int \limits ^{0} f{\left (t \right )} e^{t} \cos {\left (t \right )}\, dt\right ) \sin {\left (t \right )}\right ) e^{- t} \]