74.11.45 problem 57

Internal problem ID [16295]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.3, page 156
Problem number : 57
Date solved : Tuesday, January 28, 2025 at 09:00:53 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }&=-24 t -6-4 t \,{\mathrm e}^{-4 t}+{\mathrm e}^{-4 t} \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 15

dsolve([diff(y(t),t$2)+4*diff(y(t),t)=-24*t-6-4*t*exp(-4*t)+exp(-4*t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {t^{2} \left (-6+{\mathrm e}^{-4 t}\right )}{2} \]

✓ Solution by Mathematica

Time used: 4.082 (sec). Leaf size: 88

DSolve[{D[y[t],{t,2}]+4*D[y[t],t]==-24*t-6-4*t*Exp[-4*t]+Exp[-4*t],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to \frac {1}{2} \left (e^{-4 t}-6\right ) t^2 \\ y(t)\to \int _1^t-e^{-4 K[1]} K[1] \left (2 K[1]+6 e^{4 K[1]}-1\right )dK[1]-\int _1^0-e^{-4 K[1]} K[1] \left (2 K[1]+6 e^{4 K[1]}-1\right )dK[1] \\ \end{align*}