74.12.18 problem 18

Internal problem ID [16246]
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 : Thursday, March 13, 2025 at 08:07:08 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

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

ode:=diff(diff(y(t),t),t)-25*y(t) = 1/(1-exp(5*t)); 
dsolve(ode,y(t), singsol=all);
 

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

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

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

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

✓ Sympy. Time used: 0.322 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-25*y(t) + Derivative(y(t), (t, 2)) - 1/(1 - exp(5*t)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} + \frac {\log {\left (e^{5 t} - 1 \right )}}{50}\right ) e^{- 5 t} + \left (C_{2} + \frac {t}{10} - \frac {\log {\left (e^{5 t} - 1 \right )}}{50}\right ) e^{5 t} - \frac {1}{50} \]