11.5.12 problem 19(a)

Internal problem ID [1517]
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)
Date solved : Monday, January 27, 2025 at 04:58:40 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+2 y&=f \left (t \right ) \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.128 (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)
 

\[ y = \left (-\cos \left (t \right ) \left (\int _{0}^{t}f \left (\textit {\_z1} \right ) \sin \left (\textit {\_z1} \right ) {\mathrm e}^{\textit {\_z1}}d \textit {\_z1} \right )+\sin \left (t \right ) \left (\int _{0}^{t}f \left (\textit {\_z1} \right ) \cos \left (\textit {\_z1} \right ) {\mathrm e}^{\textit {\_z1}}d \textit {\_z1} \right )\right ) {\mathrm e}^{-t} \]

✓ Solution by Mathematica

Time used: 0.106 (sec). Leaf size: 99

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

\[ y(t)\to e^{-t} \left (-\sin (t) \int _1^0e^{K[1]} \cos (K[1]) f(K[1])dK[1]+\sin (t) \int _1^te^{K[1]} \cos (K[1]) f(K[1])dK[1]+\cos (t) \left (\int _1^t-e^{K[2]} f(K[2]) \sin (K[2])dK[2]-\int _1^0-e^{K[2]} f(K[2]) \sin (K[2])dK[2]\right )\right ) \]