Internal problem ID [5955]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 7 THE LAPLACE TRANSFORM. EXERCISES 7.5. Page 315
Problem number: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-7 y^{\prime }+6 y-{\mathrm e}^{t}-\left (\delta \left (t -2\right )\right )-\left (\delta \left (t -4\right )\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.017 (sec). Leaf size: 64
dsolve([diff(y(t),t$2)-7*diff(y(t),t)+6*y(t)=exp(t)+Dirac(t-2)+Dirac(t-4),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \relax (t ) = \frac {{\mathrm e}^{6 t}}{25}+\frac {\theta \left (t -4\right ) {\mathrm e}^{6 t -24}}{5}+\frac {\theta \left (t -2\right ) {\mathrm e}^{6 t -12}}{5}-\frac {\theta \left (t -4\right ) {\mathrm e}^{t -4}}{5}-\frac {\theta \left (t -2\right ) {\mathrm e}^{t -2}}{5}+\frac {\left (-5 t -1\right ) {\mathrm e}^{t}}{25} \]
✓ Solution by Mathematica
Time used: 0.032 (sec). Leaf size: 67
DSolve[{y''[t]-7*y'[t]+6*y[t]==Exp[t]+DiracDelta[t-2]+DiracDelta[t-4],{y[0]==9,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{25} e^{t-24} \left (5 \left (e^{5 t}-e^{20}\right ) \theta (t-4)+5 \left (e^{5 t+12}-e^{22}\right ) \theta (t-2)+e^{24} \left (-5 t-44 e^{5 t}+269\right )\right ) \\ \end{align*}