78.26.3 problem 4 (c)

Internal problem ID [18385]
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 : Thursday, March 13, 2025 at 11:55:22 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*}

✓ Maple. Time used: 0.163 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)-diff(y(t),t) = t^2; 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

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

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

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

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

✓ Sympy. Time used: 0.166 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2 - Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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