71.16.6 problem 6

Internal problem ID [14484]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 5. The Laplace Transform Method. Exercises 5.5, page 273
Problem number : 6
Date solved : Monday, March 31, 2025 at 12:28:22 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\cos \left (x \right ) \delta \left (x -\pi \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.284 (sec). Leaf size: 18

ode:=diff(diff(y(x),x),x)+4*y(x) = cos(x)*Dirac(x-Pi); 
ic:=y(0) = 0, D(y)(0) = 1; 
dsolve([ode,ic],y(x),method='laplace');
 

\[ y = -\frac {\sin \left (2 x \right ) \left (-1+\operatorname {Heaviside}\left (x -\pi \right )\right )}{2} \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 62

ode=D[y[x],{x,2}]+4*y[x]==Cos[x]*DiracDelta[x-Pi]; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{2} \left (-2 \sin (2 x) \int _1^0-\frac {1}{2} \delta (\pi -K[1])dK[1]+2 \sin (2 x) \int _1^x-\frac {1}{2} \delta (\pi -K[1])dK[1]+\sin (2 x)\right ) \]

✓ Sympy. Time used: 3.912 (sec). Leaf size: 83

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-Dirac(x - pi)*cos(x) + 4*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (- \frac {\int \operatorname {Dirac}{\left (x - \pi \right )} \sin {\left (2 x \right )} \cos {\left (x \right )}\, dx}{2} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (x - \pi \right )} \sin {\left (2 x \right )} \cos {\left (x \right )}\, dx}{2}\right ) \cos {\left (2 x \right )} + \left (\frac {\int \operatorname {Dirac}{\left (x - \pi \right )} \cos {\left (x \right )} \cos {\left (2 x \right )}\, dx}{2} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (x - \pi \right )} \cos {\left (x \right )} \cos {\left (2 x \right )}\, dx}{2} + \frac {1}{2}\right ) \sin {\left (2 x \right )} \]