14.4.13 problem 13

Internal problem ID [2531]
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 : 13
Date solved : Sunday, March 30, 2025 at 12:08:56 AM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=\left (4 y+{\mathrm e}^{-t^{2}}\right ) {\mathrm e}^{2 y} \end{align*}

With initial conditions

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

✗ Maple

ode:=diff(y(t),t) = (4*y(t)+exp(-t^2))*exp(2*y(t)); 
ic:=y(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[t],t]==(4*y[t]+Exp[-t^2])*Exp[2*y[t]]; 
ic={y[0]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((-4*y(t) - exp(-t**2))*exp(2*y(t)) + Derivative(y(t), t),0) 
ics = {y(0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

NotImplementedError : The given ODE -(4*y(t)*exp(t**2) + 1)*exp(-t**2 + 2*y(t)) + Derivative(y(t), t) cannot be solved by the factorable group method