68.11.10 problem 22

Internal problem ID [17547]
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 : 22
Date solved : Thursday, October 02, 2025 at 02:25:18 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=3 t \,{\mathrm e}^{-t} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 28

ode:=diff(diff(y(t),t),t)+4*y(t) = 3*t*exp(-t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \sin \left (2 t \right ) c_2 +\cos \left (2 t \right ) c_1 +\frac {3 \left (5 t +2\right ) {\mathrm e}^{-t}}{25} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 34

ode=D[y[t],{t,2}]+4*y[t]==3*t*Exp[-t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {3}{25} e^{-t} (5 t+2)+c_1 \cos (2 t)+c_2 \sin (2 t) \end{align*}

✓ Sympy. Time used: 0.058 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-3*t*exp(-t) + 4*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} \sin {\left (2 t \right )} + C_{2} \cos {\left (2 t \right )} + \frac {3 t e^{- t}}{5} + \frac {6 e^{- t}}{25} \]