14.17.4 problem 22

Internal problem ID [2682]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.10, Some useful properties of Laplace transform. Excercises page 238
Problem number : 22
Date solved : Tuesday, March 04, 2025 at 02:34:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+7 y&=\sin \left (t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.995 (sec). Leaf size: 36

ode:=diff(diff(y(t),t),t)-2*diff(y(t),t)+7*y(t) = sin(t); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -\frac {{\mathrm e}^{t} \sqrt {6}\, \sin \left (\sqrt {6}\, t \right )}{60}-\frac {{\mathrm e}^{t} \cos \left (\sqrt {6}\, t \right )}{20}+\frac {3 \sin \left (t \right )}{20}+\frac {\cos \left (t \right )}{20} \]

✓ Mathematica. Time used: 0.869 (sec). Leaf size: 49

ode=D[y[t],{t,2}]-2*D[y[t],t]+7*y[t]==Sin[t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{60} \left (9 \sin (t)-\sqrt {6} e^t \sin \left (\sqrt {6} t\right )+3 \cos (t)-3 e^t \cos \left (\sqrt {6} t\right )\right ) \]

✓ Sympy. Time used: 0.412 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(7*y(t) - sin(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)
 

\[ y{\left (t \right )} = \left (- \frac {\sqrt {6} \sin {\left (\sqrt {6} t \right )}}{60} - \frac {\cos {\left (\sqrt {6} t \right )}}{20}\right ) e^{t} + \frac {3 \sin {\left (t \right )}}{20} + \frac {\cos {\left (t \right )}}{20} \]