68.8.32 problem 32

Internal problem ID [17455]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Review exercises, page 80
Problem number : 32
Date solved : Thursday, October 02, 2025 at 02:23:31 PM
CAS classification : [_exact, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} {\mathrm e}^{y t} y-2 t +t \,{\mathrm e}^{y t} y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.189 (sec). Leaf size: 14

ode:=y(t)*exp(t*y(t))-2*t+t*exp(t*y(t))*diff(y(t),t) = 0; 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {\ln \left (t^{2}+1\right )}{t} \]

✓ Mathematica. Time used: 0.309 (sec). Leaf size: 15

ode=(y[t]*Exp[t*y[t]]-2*t)+(t*Exp[t*y[t]])*D[y[t],t]==0; 
ic={y[0]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(t), t) - 2*exp(-t*y(t)) + y(t)/t cannot be solved by the factorable group method