50.24.3 problem 1(c)

Internal problem ID [8169]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 7. Laplace Transforms. Section 7.5 The Unit Step and Impulse Functions. Page 303
Problem number : 1(c)
Date solved : Wednesday, March 05, 2025 at 05:31:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-y&=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: 0.520 (sec). Leaf size: 15

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

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

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 20

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

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

✓ Sympy. Time used: 0.103 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2 - y(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)
 

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