66.16.25 problem 26

Internal problem ID [16208]
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 : 26
Date solved : Thursday, October 02, 2025 at 10:43:36 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y&=2 \,{\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: 0.042 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+4*y(t) = 2*exp(-2*t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {\sin \left (2 t \right )}{4}-\frac {\cos \left (2 t \right )}{4}+\frac {{\mathrm e}^{-2 t}}{4} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 25

ode=D[y[t],{t,2}]+4*y[t]==2*Exp[-2*t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{4} \left (e^{-2 t}+\sin (2 t)-\cos (2 t)\right ) \end{align*}

✓ Sympy. Time used: 0.065 (sec). Leaf size: 24

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

\[ y{\left (t \right )} = \frac {\sin {\left (2 t \right )}}{4} - \frac {\cos {\left (2 t \right )}}{4} + \frac {e^{- 2 t}}{4} \]