68.11.42 problem 54

Internal problem ID [17579]
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 : Thursday, October 02, 2025 at 02:25:40 PM
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*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 25

ode:=diff(diff(y(t),t),t)-3*diff(y(t),t) = -exp(3*t)-2*t; 
ic:=[y(0) = 0, D(y)(0) = 8/9]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 2.601 (sec). Leaf size: 132

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

\begin{align*} 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] \end{align*}

✓ Sympy. Time used: 0.138 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(2*t + exp(3*t) - 3*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 8/9} 
dsolve(ode,func=y(t),ics=ics)
 

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