68.12.30 problem 30

Internal problem ID [17624]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number : 30
Date solved : Thursday, October 02, 2025 at 02:26:23 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+8 y^{\prime }+16 y&=\frac {{\mathrm e}^{-4 t}}{t^{2}+1} \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)+8*diff(y(t),t)+16*y(t) = exp(-4*t)/(t^2+1); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{-4 t} \left (c_2 +t c_1 -\frac {\ln \left (t^{2}+1\right )}{2}+\arctan \left (t \right ) t \right ) \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 50

ode=D[y[t],{t,2}]+8*D[y[t],t]+16*y[t]==Exp[-4*t]*1/(1+t^2); 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{2} e^{-4 t} \left (2 t \int _1^t\frac {1}{K[1]^2+1}dK[1]-\log \left (t^2+1\right )+2 (c_2 t+c_1)\right ) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(16*y(t) + 8*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - exp(-4*t)/(t**2 + 1),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(t), t) - (-16*t**2*y(t)*exp(4*t) - t**2*exp(4*t)*Derivative(y(t), (t, 2)) - 16*y(t)*exp(4*t) - exp(4*t)*Derivative(y(t), (t, 2)) + 1)*exp(-4*t)/(8*(t**2 + 1)) cannot be solved by the factorable group method