78.26.4 problem 5 (a)

Internal problem ID [18465]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 9. Laplace transforms. Section 53. More about Convolutions. The Unit Step and Impulse Functions. Problems at page 481
Problem number : 5 (a)
Date solved : Tuesday, January 28, 2025 at 11:50:28 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=f \left (t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 0.307 (sec). Leaf size: 39

dsolve([diff(y(t),t$2)+3*diff(y(t),t)+2*y(t)=f(t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = -\int _{0}^{t}{\mathrm e}^{-2 t +2 \textit {\_U1}} f \left (\textit {\_U1} \right )d \textit {\_U1} +\int _{0}^{t}{\mathrm e}^{-t +\textit {\_U1}} f \left (\textit {\_U1} \right )d \textit {\_U1} \]

✓ Solution by Mathematica

Time used: 0.048 (sec). Leaf size: 87

DSolve[{D[y[t],{t,2}]+3*D[y[t],t]+2*y[t]==f[t],{y[0]==0,Derivative[1][y][0] == 0}},y[t],t,IncludeSingularSolutions -> True]
 

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