71.16.7 problem 7

Internal problem ID [14477]
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 : 7
Date solved : Thursday, March 13, 2025 at 03:31:16 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+a^{2} y&=\delta \left (x -\pi \right ) f \left (x \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: 10.466 (sec). Leaf size: 24

ode:=diff(diff(y(x),x),x)+a^2*y(x) = Dirac(x-Pi)*f(x); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(x),method='laplace');
 

\[ y = \frac {\operatorname {Heaviside}\left (x -\pi \right ) f \left (\pi \right ) \sin \left (a \left (x -\pi \right )\right )}{a} \]

✓ Mathematica. Time used: 0.102 (sec). Leaf size: 126

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

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

✓ Sympy. Time used: 1.450 (sec). Leaf size: 104

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

\[ y{\left (x \right )} = \left (- \frac {i \int \operatorname {Dirac}{\left (x - \pi \right )} f{\left (x \right )} e^{- i a x}\, dx}{2 a} + \frac {i \int \limits ^{0} \operatorname {Dirac}{\left (x - \pi \right )} f{\left (x \right )} e^{- i a x}\, dx}{2 a}\right ) e^{i a x} + \left (\frac {i \int \operatorname {Dirac}{\left (x - \pi \right )} f{\left (x \right )} e^{i a x}\, dx}{2 a} - \frac {i \int \limits ^{0} \operatorname {Dirac}{\left (x - \pi \right )} f{\left (x \right )} e^{i a x}\, dx}{2 a}\right ) e^{- i a x} \]