66.16.39 problem 41

Internal problem ID [16222]
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 : 41
Date solved : Thursday, October 02, 2025 at 10:43:48 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y&=t +{\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: 0.020 (sec). Leaf size: 26

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

\[ y = -\frac {\sin \left (2 t \right )}{40}-\frac {\cos \left (2 t \right )}{5}+\frac {t}{4}+\frac {{\mathrm e}^{-t}}{5} \]

✓ Mathematica. Time used: 0.237 (sec). Leaf size: 147

ode=D[y[t],{t,2}]+4*y[t]==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 (2 t) \int _1^0\frac {1}{2} e^{-K[2]} \cos (2 K[2]) \left (e^{K[2]} K[2]+1\right )dK[2]+\sin (2 t) \int _1^t\frac {1}{2} e^{-K[2]} \cos (2 K[2]) \left (e^{K[2]} K[2]+1\right )dK[2]+\cos (2 t) \left (\int _1^t-e^{-K[1]} \cos (K[1]) \left (e^{K[1]} K[1]+1\right ) \sin (K[1])dK[1]-\int _1^0-e^{-K[1]} \cos (K[1]) \left (e^{K[1]} K[1]+1\right ) \sin (K[1])dK[1]\right ) \end{align*}

✓ Sympy. Time used: 0.067 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t + 4*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)
 

\[ y{\left (t \right )} = \frac {t}{4} - \frac {\sin {\left (2 t \right )}}{40} - \frac {\cos {\left (2 t \right )}}{5} + \frac {e^{- t}}{5} \]