20.21.24 problem Problem 24

Internal problem ID [3951]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.4. page 689
Problem number : Problem 24
Date solved : Tuesday, March 04, 2025 at 05:19:57 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }+2 y&=3 \cos \left (t \right )+\sin \left (t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 2.731 (sec). Leaf size: 23

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

\[ y = \frac {3 \cos \left (t \right )}{5}-\frac {4 \sin \left (t \right )}{5}-{\mathrm e}^{t}+\frac {7 \,{\mathrm e}^{2 t}}{5} \]

✓ Mathematica. Time used: 0.11 (sec). Leaf size: 29

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

\[ y(t)\to \frac {1}{5} \left (e^t \left (7 e^t-5\right )-4 \sin (t)+3 \cos (t)\right ) \]

✓ Sympy. Time used: 0.231 (sec). Leaf size: 27

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

\[ y{\left (t \right )} = \frac {7 e^{2 t}}{5} - e^{t} - \frac {4 \sin {\left (t \right )}}{5} + \frac {3 \cos {\left (t \right )}}{5} \]