72.23.5 problem 3 (e)

Internal problem ID [19742]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 9. Laplace transforms. Section 50. Applications to differential equations. Problems at page 462
Problem number : 3 (e)
Date solved : Thursday, October 02, 2025 at 04:41:51 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 5 y+2 y^{\prime }+y^{\prime \prime }&=3 \,{\mathrm e}^{-x} \sin \left (x \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.187 (sec). Leaf size: 16

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)+5*y(x) = 3*exp(-x)*sin(x); 
ic:=[y(0) = 0, D(y)(0) = 3]; 
dsolve([ode,op(ic)],y(x),method='laplace');
 

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

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 18

ode=D[y[x],{x,2}]+2*D[y[x],x]+5*y[x]==3*Exp[-x]*Sin[x]; 
ic={y[0]==0,Derivative[1][y][0] == 3}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-x} (\sin (x)+\sin (2 x)) \end{align*}

✓ Sympy. Time used: 0.176 (sec). Leaf size: 14

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

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