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 : Monday, January 27, 2025 at 03:45:10 PM
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*}

✓ Solution by Maple

Time used: 0.520 (sec). Leaf size: 15

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

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

✓ Solution by Mathematica

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

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

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