74.18.22 problem 28

Internal problem ID [16579]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Chapter 4 review exercises, page 219
Problem number : 28
Date solved : Tuesday, January 28, 2025 at 09:11:28 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-9 y&=\frac {1}{1+{\mathrm e}^{3 t}} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 69

dsolve(diff(y(t),t$2)-9*y(t)=1/(1+exp(3*t)),y(t), singsol=all)
 

\[ y = \frac {\left (\ln \left ({\mathrm e}^{2 t}-{\mathrm e}^{t}+1\right ) {\mathrm e}^{6 t}-\ln \left (1+{\mathrm e}^{3 t}\right )+\ln \left (1+{\mathrm e}^{t}\right ) {\mathrm e}^{6 t}+\left (18 c_{1} -3 \ln \left ({\mathrm e}^{t}\right )\right ) {\mathrm e}^{6 t}+18 c_{2} -{\mathrm e}^{3 t}\right ) {\mathrm e}^{-3 t}}{18} \]

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 63

DSolve[D[y[t],{t,2}]-9*y[t]==1/(1+Exp[3*t]),y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{9} e^{3 t} \text {arctanh}\left (2 e^{3 t}+1\right )-\frac {1}{18} e^{-3 t} \log \left (6 \left (e^{3 t}+1\right )\right )+c_1 e^{3 t}+c_2 e^{-3 t}-\frac {1}{18} \]