78.26.3 problem 4 (c)

Internal problem ID [18464]
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 : 4 (c)
Date solved : Tuesday, January 28, 2025 at 11:50:27 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }-y^{\prime }&=t^{2} \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: 0.163 (sec). Leaf size: 23

dsolve([diff(y(t),t$2)-diff(y(t),t)=t^2,y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = -t^{2}-\frac {t^{3}}{3}-2 t +2 \,{\mathrm e}^{t}-2 \]

✓ Solution by Mathematica

Time used: 0.045 (sec). Leaf size: 27

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

\[ y(t)\to -\frac {t^3}{3}-t^2-2 t+2 e^t-2 \]