Internal problem ID [867]
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: 19(a).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+2 y^{\prime }+2 y-f \relax (t )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 43
dsolve([diff(y(t),t$2)+2*diff(y(t),t)+2*y(t)=f(t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \relax (t ) = \left (\left (\int _{0}^{t}f \left (\textit {\_z1} \right ) \cos \left (\textit {\_z1} \right ) {\mathrm e}^{\textit {\_z1}}d \textit {\_z1} \right ) \sin \relax (t )-\cos \relax (t ) \left (\int _{0}^{t}f \left (\textit {\_z1} \right ) \sin \left (\textit {\_z1} \right ) {\mathrm e}^{\textit {\_z1}}d \textit {\_z1} \right )\right ) {\mathrm e}^{-t} \]
✓ Solution by Mathematica
Time used: 0.055 (sec). Leaf size: 98
DSolve[{y''[t]+2*y'[t]+2*y[t]==f[t],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to e^{-t} \left (\sin (t) \left (\int _1^te^{K[1]} \cos (K[1]) f(K[1])dK[1]-\int _1^0e^{K[1]} \cos (K[1]) f(K[1])dK[1]\right )+\cos (t) \left (\int _1^t-e^{K[2]} f(K[2]) \sin (K[2])dK[2]-\int _1^0-e^{K[2]} f(K[2]) \sin (K[2])dK[2]\right )\right ) \\ \end{align*}