68.11.32 problem 44

Internal problem ID [17569]
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 : 44
Date solved : Thursday, October 02, 2025 at 02:25:32 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 33

ode:=diff(diff(y(t),t),t)-3*diff(y(t),t) = t^2-exp(3*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\left (1-3 t +3 c_1 \right ) {\mathrm e}^{3 t}}{9}-\frac {t^{3}}{9}-\frac {t^{2}}{9}-\frac {2 t}{27}+c_2 \]

✓ Mathematica. Time used: 2.512 (sec). Leaf size: 46

ode=D[y[t],{t,2}]-3*D[y[t],t]==t^2-Exp[3*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \int _1^te^{3 K[2]} \left (c_1+\int _1^{K[2]}\left (e^{-3 K[1]} K[1]^2-1\right )dK[1]\right )dK[2]+c_2 \end{align*}

✓ Sympy. Time used: 0.132 (sec). Leaf size: 29

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

\[ y{\left (t \right )} = C_{1} - \frac {t^{3}}{9} - \frac {t^{2}}{9} - \frac {2 t}{27} + \left (C_{2} - \frac {t}{3}\right ) e^{3 t} \]