72.16.10 problem 10

Internal problem ID [14827]
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 : 10
Date solved : Thursday, March 13, 2025 at 04:20:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+7 y^{\prime }+12 y&=3 \,{\mathrm e}^{-t} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+7*diff(y(t),t)+12*y(t) = 3*exp(-t); 
ic:=y(0) = 2, D(y)(0) = 1; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \frac {15 \,{\mathrm e}^{-3 t}}{2}-6 \,{\mathrm e}^{-4 t}+\frac {{\mathrm e}^{-t}}{2} \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 26

ode=D[y[t],{t,2}]+7*D[y[t],t]+12*y[t]==3*Exp[-t]; 
ic={y[0]==2,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{2} e^{-4 t} \left (15 e^t+e^{3 t}-12\right ) \]

✓ Sympy. Time used: 0.269 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(12*y(t) + 7*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 3*exp(-t),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {1}{2} + \frac {15 e^{- 2 t}}{2} - 6 e^{- 3 t}\right ) e^{- t} \]