80.7.37 problem D 6

Internal problem ID [21356]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 7. System of first order equations. Excercise 7.6 at page 162
Problem number : D 6
Date solved : Sunday, October 12, 2025 at 05:51:28 AM
CAS classification : system_of_ODEs

\begin{align*} t \left (\frac {d}{d t}x \left (t \right )\right )+\frac {d}{d t}y \left (t \right )&=1\\ \frac {d}{d t}y \left (t \right )+x \left (t \right )+{\mathrm e}^{\frac {d}{d t}x \left (t \right )}&=1 \end{align*}

✓ Maple. Time used: 0.057 (sec). Leaf size: 40

ode:=[t*diff(x(t),t)+diff(y(t),t) = 1, diff(y(t),t)+x(t)+exp(diff(x(t),t)) = 1]; 
dsolve(ode);
 

\begin{align*} \{x \left (t \right ) &= t c_2 -{\mathrm e}^{c_2}, x \left (t \right ) = t \ln \left (t \right )-t\} \\ \{y \left (t \right ) &= \int -t \left (\frac {d}{d t}x \left (t \right )\right )d t +t +c_1\} \\ \end{align*}

✓ Mathematica. Time used: 0.186 (sec). Leaf size: 77

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

\begin{align*} x(t)&\to (c_1-i \pi ) t+e^{c_1}\\ y(t)&\to \int _1^t\left (-c_1 K[1]+W\left (\frac {e^{c_1+\frac {e^{c_1}}{K[1]}}}{K[1]}\right ) K[1]+i \pi K[1]-e^{c_1}+1\right )dK[1]+c_2 \end{align*}

✗ Sympy

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

NotImplementedError :