74.7.51 problem 54

Internal problem ID [16057]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.5, page 64
Problem number : 54
Date solved : Monday, March 31, 2025 at 02:37:12 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} 1+y-t y^{\prime }&=\ln \left (y^{\prime }\right ) \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 22

ode:=1+y(t)-t*diff(y(t),t) = ln(diff(y(t),t)); 
dsolve(ode,y(t), singsol=all);
 

\begin{align*} y &= \ln \left (-\frac {1}{t}\right )-2 \\ y &= -1+t c_1 +\ln \left (c_1 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 26

ode=1+y[t]-t*D[y[t],t]==Log[D[y[t],t]]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)\to c_1 t-1+\log (c_1) \\ y(t)\to \log \left (-\frac {1}{t}\right )-2 \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(t), t) - LambertW(t*exp(y(t) + 1))/t cannot be solved by the factorable group method