Internal problem ID [14850]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_y]]
\[ \boxed {y^{\prime \prime }-2 y^{\prime }=\frac {1}{1+{\mathrm e}^{2 t}}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 40
dsolve(diff(y(t),t$2)-2*diff(y(t),t)=1/(1+exp(2*t)),y(t), singsol=all)
\[ y \left (t \right ) = \frac {\ln \left ({\mathrm e}^{2 t}+1\right ) \left ({\mathrm e}^{2 t}+1\right )}{4}+\frac {\left (2 c_{1} -2 \ln \left ({\mathrm e}^{t}\right )\right ) {\mathrm e}^{2 t}}{4}-\frac {t}{2}+c_{2} -\frac {1}{4} \]
✓ Solution by Mathematica
Time used: 0.15 (sec). Leaf size: 61
DSolve[y''[t]-2*y'[t]==1/(1+Exp[2*t]),y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {1}{8} \left (\left (4 e^{2 t}+2\right ) \text {arctanh}\left (2 e^{2 t}+1\right )-4 t+\log \left (-4 e^{2 t} \left (e^{2 t}+1\right )\right )+4 c_1 e^{2 t}+8 c_2\right ) \]