76.15.8 problem 8

Internal problem ID [17570]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.5 (Nonhomogeneous Equations, Method of Undetermined Coefficients). Problems at page 260
Problem number : 8
Date solved : Thursday, March 13, 2025 at 10:13:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 32

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

\[ y(t)\to \frac {1}{9} e^{-t} \left (-6 t+9 c_2 e^{3 t}-2+9 c_1\right ) \]

✓ Sympy. Time used: 0.195 (sec). Leaf size: 19

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

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