71.16.5 problem 5

Internal problem ID [14483]
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 : 5
Date solved : Monday, March 31, 2025 at 12:28:20 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+5 y&=\cos \left (x \right )+2 \delta \left (x -\pi \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.323 (sec). Leaf size: 47

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)+5*y(x) = cos(x)+2*Dirac(x-Pi); 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(x),method='laplace');
 

\[ y = -\frac {\sin \left (x \right )}{10}+\frac {\cos \left (x \right )}{5}+\frac {\left (16 \cos \left (2 x \right )-7 \sin \left (2 x \right )\right ) {\mathrm e}^{x}}{20}+\sin \left (2 x \right ) \operatorname {Heaviside}\left (x -\pi \right ) {\mathrm e}^{x -\pi } \]

✓ Mathematica. Time used: 0.3 (sec). Leaf size: 185

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*Dirac(x - pi) + 5*y(x) - cos(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out