71.16.5 problem 5

Internal problem ID [14554]
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 : Tuesday, January 28, 2025 at 06:43:23 AM
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*}

✓ Solution by Maple

Time used: 10.450 (sec). Leaf size: 46

dsolve([diff(y(x),x$2)-2*diff(y(x),x)+5*y(x)=cos(x)+2*Dirac(x-Pi),y(0) = 1, D(y)(0) = 0],y(x), singsol=all)
 

\[ y = \sin \left (2 x \right ) \operatorname {Heaviside}\left (x -\pi \right ) {\mathrm e}^{x -\pi }+\frac {4 \,{\mathrm e}^{x} \cos \left (2 x \right )}{5}-\frac {7 \,{\mathrm e}^{x} \sin \left (2 x \right )}{20}+\frac {\cos \left (x \right )}{5}-\frac {\sin \left (x \right )}{10} \]

✓ Solution by Mathematica

Time used: 0.300 (sec). Leaf size: 185

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

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