Internal problem ID [865]
Book: Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima,
Meade
Section: Chapter 6.5, The Laplace Transform. Impulse functions. page 273
Problem number: 10(c).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {y^{\prime }}{4}+y-\left (\delta \left (t -1\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: 0.019 (sec). Leaf size: 28
dsolve([diff(y(t),t$2)+1/4*diff(y(t),t)+y(t)=Dirac(t-1),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \relax (t ) = \frac {8 \theta \left (t -1\right ) \sqrt {7}\, {\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, \left (t -1\right )}{8}\right )}{21} \]
✓ Solution by Mathematica
Time used: 0.032 (sec). Leaf size: 42
DSolve[{y''[t]+1/4*y'[t]+y[t]==DiracDelta[t-1],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {8 e^{\frac {1}{8}-\frac {t}{8}} \theta (t-1) \sin \left (\frac {3}{8} \sqrt {7} (t-1)\right )}{3 \sqrt {7}} \\ \end{align*}