76.15.21 problem 22

Internal problem ID [17583]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.5 (Nonhomogeneous Equations, Method of Undetermined Coefficients). Problems at page 260
Problem number : 22
Date solved : Thursday, March 13, 2025 at 10:14:38 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=4 \,{\mathrm e}^{-t} \cos \left (2 t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 28

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+5*y(t) = 4*exp(-t)*cos(2*t); 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.075 (sec). Leaf size: 31

ode=D[y[t],{t,2}]+2*D[y[t],t]+5*y[t]==4*Exp[-t]*Cos[2*t]; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{2} e^{-t} ((2 t+1) \sin (2 t)+2 \cos (2 t)) \]

✓ Sympy. Time used: 0.315 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 4*exp(-t)*cos(2*t),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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