71.14.7 problem 13

Internal problem ID [14470]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 5. The Laplace Transform Method. Exercises 5.3, page 255
Problem number : 13
Date solved : Monday, March 31, 2025 at 12:27:53 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-2\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.104 (sec). Leaf size: 15

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

\[ y = \left (3 x -3\right ) {\mathrm e}^{x}+\cos \left (x \right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 16

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

\[ y(x)\to 3 e^x (x-1)+\cos (x) \]

✓ Sympy. Time used: 0.192 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = \left (3 x - 3\right ) e^{x} + \cos {\left (x \right )} \]