14.4.9 problem 9

Internal problem ID [2527]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 1. First order differential equations. Section 1.10. Existence-uniqueness theorem. Excercises page 80
Problem number : 9
Date solved : Sunday, March 30, 2025 at 12:08:46 AM
CAS classification : [_Riccati]

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

With initial conditions

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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