74.12.18 problem 18

Internal problem ID [16325]
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 : 18
Date solved : Tuesday, January 28, 2025 at 09:04:43 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-25 y&=\frac {1}{1-{\mathrm e}^{5 t}} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 77

dsolve(diff(y(t),t$2)-25*y(t)=(1-exp(5*t))^(-1),y(t), singsol=all)
 

\[ y = -\frac {\left (\ln \left ({\mathrm e}^{4 t}+{\mathrm e}^{3 t}+{\mathrm e}^{2 t}+{\mathrm e}^{t}+1\right ) {\mathrm e}^{10 t}+\ln \left ({\mathrm e}^{t}-1\right ) {\mathrm e}^{10 t}-5 \ln \left ({\mathrm e}^{t}\right ) {\mathrm e}^{10 t}-50 c_{2} {\mathrm e}^{10 t}+{\mathrm e}^{5 t}-\ln \left (1-{\mathrm e}^{5 t}\right )-50 c_{1} \right ) {\mathrm e}^{-5 t}}{50} \]

✓ Solution by Mathematica

Time used: 0.172 (sec). Leaf size: 60

DSolve[D[y[t],{t,2}]-25*y[t]==(1-Exp[5*t])^(-1),y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{50} \left (-2 e^{5 t} \text {arctanh}\left (1-2 e^{5 t}\right )+e^{-5 t} \log \left (e^{5 t}-1\right )+50 c_1 e^{5 t}+50 c_2 e^{-5 t}-1\right ) \]