71.16.2 problem 2

Internal problem ID [14551]
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 : 2
Date solved : Tuesday, January 28, 2025 at 06:43:20 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }-3 y&=\delta \left (x -1\right )+2 \operatorname {Heaviside}\left (x -2\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 10.280 (sec). Leaf size: 34

dsolve([diff(y(x),x)-3*y(x)=Dirac(x-1)+2*Heaviside(x-2),y(0) = 0],y(x), singsol=all)
 

\[ y = -\frac {2 \operatorname {Heaviside}\left (x -2\right )}{3}+\frac {2 \operatorname {Heaviside}\left (x -2\right ) {\mathrm e}^{3 x -6}}{3}+\operatorname {Heaviside}\left (x -1\right ) {\mathrm e}^{3 x -3} \]

✓ Solution by Mathematica

Time used: 0.686 (sec). Leaf size: 134

DSolve[{D[y[x],x]-3*y[x]==DiracDelta[x-1]+2*UnitStep[x-2],{y[0]==0}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \theta (2-x) \left (e^{3 x} \int _0^x\frac {\delta (K[1]-1)}{e^3}dK[1]-\frac {1}{3} e^{3 x-6} \left (3 e^6 \int _1^xe^{-3 K[2]} (\delta (K[2]-1)+2)dK[2]-3 e^3 \theta (0)+e^3+2\right )\right )+\frac {1}{3} e^{3 x-6} \left (3 e^6 \int _1^xe^{-3 K[2]} (\delta (K[2]-1)+2)dK[2]-3 e^3 \theta (0)+e^3+2\right ) \]