66.16.24 problem 25

Internal problem ID [16207]
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 : 25
Date solved : Thursday, October 02, 2025 at 10:43:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+9 y&={\mathrm e}^{-t} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.044 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+9*y(t) = exp(-t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {\sin \left (3 t \right )}{30}-\frac {\cos \left (3 t \right )}{10}+\frac {{\mathrm e}^{-t}}{10} \]

✓ Mathematica. Time used: 0.09 (sec). Leaf size: 113

ode=D[y[t],{t,2}]+9*y[t]==Exp[-t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\sin (3 t) \int _1^0\frac {1}{3} e^{-K[2]} \cos (3 K[2])dK[2]+\sin (3 t) \int _1^t\frac {1}{3} e^{-K[2]} \cos (3 K[2])dK[2]+\cos (3 t) \left (\int _1^t-\frac {1}{3} e^{-K[1]} \sin (3 K[1])dK[1]-\int _1^0-\frac {1}{3} e^{-K[1]} \sin (3 K[1])dK[1]\right ) \end{align*}

✓ Sympy. Time used: 0.057 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(9*y(t) + Derivative(y(t), (t, 2)) - exp(-t),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {\sin {\left (3 t \right )}}{30} - \frac {\cos {\left (3 t \right )}}{10} + \frac {e^{- t}}{10} \]