74.18.23 problem 29

Internal problem ID [16580]
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 : 29
Date solved : Tuesday, January 28, 2025 at 09:11:29 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 40

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

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

✓ Solution by Mathematica

Time used: 0.160 (sec). Leaf size: 61

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

\[ y(t)\to \frac {1}{8} \left (\left (4 e^{2 t}+2\right ) \text {arctanh}\left (2 e^{2 t}+1\right )-4 t+\log \left (-4 e^{2 t} \left (e^{2 t}+1\right )\right )+4 c_1 e^{2 t}+8 c_2\right ) \]