74.11.46 problem 58

Internal problem ID [16296]
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 : 58
Date solved : Tuesday, January 28, 2025 at 09:00:56 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }&={\mathrm e}^{-3 t}-{\mathrm e}^{3 t} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 23

dsolve([diff(y(t),t$2)-3*diff(y(t),t)=exp(-3*t)-exp(3*t),y(0) = 1, D(y)(0) = 1],y(t), singsol=all)
 

\[ y = \frac {4}{9}+\frac {\left (3-2 t \right ) {\mathrm e}^{3 t}}{6}+\frac {{\mathrm e}^{-3 t}}{18} \]

✓ Solution by Mathematica

Time used: 3.128 (sec). Leaf size: 73

DSolve[{D[y[t],{t,2}]-3*D[y[t],t]==Exp[-3*t]-Exp[3*t],{y[0]==1,Derivative[1][y][0] ==1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \int _1^t\frac {1}{6} e^{-3 K[1]} \left (e^{6 K[1]} (7-6 K[1])-1\right )dK[1]-\int _1^0\frac {1}{6} e^{-3 K[1]} \left (e^{6 K[1]} (7-6 K[1])-1\right )dK[1]+1 \]