74.11.42 problem 54

Internal problem ID [16292]
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 : 54
Date solved : Tuesday, January 28, 2025 at 09:00:45 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }&=-{\mathrm e}^{3 t}-2 t \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&={\frac {8}{9}} \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 25

dsolve([diff(y(t),t$2)-3*diff(y(t),t)=-exp(3*t)-2*t,y(0) = 0, D(y)(0) = 8/9],y(t), singsol=all)
 

\[ y = \frac {\left (-3 t +3\right ) {\mathrm e}^{3 t}}{9}+\frac {t^{2}}{3}+\frac {2 t}{9}-\frac {1}{3} \]

✓ Solution by Mathematica

Time used: 4.437 (sec). Leaf size: 132

DSolve[{D[y[t],{t,2}]-3*D[y[t],t]==-Exp[3*t]-2*t,{y[0]==0,Derivative[1][y][0] ==8/9}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \int _1^t\frac {1}{9} e^{3 K[2]} \left (-9 \int _1^0\left (-2 e^{-3 K[1]} K[1]-1\right )dK[1]+9 \int _1^{K[2]}\left (-2 e^{-3 K[1]} K[1]-1\right )dK[1]+8\right )dK[2]-\int _1^0\frac {1}{9} e^{3 K[2]} \left (-9 \int _1^0\left (-2 e^{-3 K[1]} K[1]-1\right )dK[1]+9 \int _1^{K[2]}\left (-2 e^{-3 K[1]} K[1]-1\right )dK[1]+8\right )dK[2] \]