72.16.12 problem 12

Internal problem ID [14829]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 4. Forcing and Resonance. Section 4.1 page 399
Problem number : 12
Date solved : Thursday, March 13, 2025 at 04:20:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+7 y^{\prime }+10 y&={\mathrm e}^{-2 t} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 22

ode:=diff(diff(y(t),t),t)+7*diff(y(t),t)+10*y(t) = exp(-2*t); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \frac {\left (-1+3 t \right ) {\mathrm e}^{-2 t}}{9}+\frac {{\mathrm e}^{-5 t}}{9} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 27

ode=D[y[t],{t,2}]+7*D[y[t],t]+10*y[t]==Exp[-2*t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{9} e^{-5 t} \left (e^{3 t} (3 t-1)+1\right ) \]

✓ Sympy. Time used: 0.316 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(10*y(t) + 7*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - exp(-2*t),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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