74.12.30 problem 30

Internal problem ID [16337]
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 : Tuesday, January 28, 2025 at 09:05:08 AM
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*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 26

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

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

✓ Solution by Mathematica

Time used: 0.040 (sec). Leaf size: 50

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

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