8.5.4 problem 4

Internal problem ID [2541]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 1. First order differential equations. Section 1.17. What to do in practice. Excercises page 126
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 05:46:52 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Bernoulli]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 8

ode:=diff(y(t),t) = y(t)^2*exp(t)-2*y(t); 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = {\mathrm e}^{-t} \]

✓ Mathematica. Time used: 0.145 (sec). Leaf size: 10

ode=D[y[t],t]==Exp[t]*y[t]^2-2*y[t]; 
ic={y[0]==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-t} \end{align*}

✓ Sympy. Time used: 0.148 (sec). Leaf size: 7

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

\[ y{\left (t \right )} = e^{- t} \]