82.4.20 problem 28-20

Internal problem ID [21838]
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 : 28-20
Date solved : Thursday, October 02, 2025 at 08:02:44 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }+4 y^{\prime \prime }+5 y^{\prime }+2 y&=10 \cos \left (t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.113 (sec). Leaf size: 29

ode:=diff(diff(diff(y(t),t),t),t)+4*diff(diff(y(t),t),t)+5*diff(y(t),t)+2*y(t) = 10*cos(t); 
ic:=[y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 3]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 30

ode=D[y[t],{t,3}]+4*D[y[t],{t,2}]+5*D[y[t],t]+2*y[t]==10*Cos[t]; 
ic={y[0]==0,Derivative[1][y][0] ==0,Derivative[2][y][0] ==3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.180 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(2*y(t) - 10*cos(t) + 5*Derivative(y(t), t) + 4*Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 3)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0, Subs(Derivative(y(t), (t, 2)), t, 0): 3} 
dsolve(ode,func=y(t),ics=ics)
 

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