71.16.4 problem 4

Internal problem ID [14553]
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 : 4
Date solved : Tuesday, January 28, 2025 at 06:43:22 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+y&=2 \delta \left (x -1\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 10.121 (sec). Leaf size: 22

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

\[ y = 2 \operatorname {Heaviside}\left (x -1\right ) {\mathrm e}^{x -1} \left (x -1\right )+{\mathrm e}^{x} x \]

✓ Solution by Mathematica

Time used: 0.027 (sec). Leaf size: 89

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

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