14.4.3 problem 3

Internal problem ID [2521]
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 : 3
Date solved : Sunday, March 30, 2025 at 12:08:01 AM
CAS classification : [_Riccati]

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

With initial conditions

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

✓ Maple. Time used: 0.219 (sec). Leaf size: 63

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

\[ y = \frac {\left (\operatorname {BesselJ}\left (1, 2\right ) \operatorname {BesselY}\left (1, 2 \,{\mathrm e}^{\frac {t}{2}}\right )-\operatorname {BesselJ}\left (1, 2 \,{\mathrm e}^{\frac {t}{2}}\right ) \operatorname {BesselY}\left (1, 2\right )\right ) {\mathrm e}^{\frac {t}{2}}}{\operatorname {BesselJ}\left (1, 2\right ) \operatorname {BesselY}\left (0, 2 \,{\mathrm e}^{\frac {t}{2}}\right )-\operatorname {BesselJ}\left (0, 2 \,{\mathrm e}^{\frac {t}{2}}\right ) \operatorname {BesselY}\left (1, 2\right )} \]

✓ Mathematica. Time used: 0.282 (sec). Leaf size: 79

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

\[ y(t)\to \frac {\sqrt {e^t} \left (\operatorname {BesselJ}(1,2) \operatorname {BesselY}\left (1,2 \sqrt {e^t}\right )-\operatorname {BesselY}(1,2) \operatorname {BesselJ}\left (1,2 \sqrt {e^t}\right )\right )}{\operatorname {BesselJ}(1,2) \operatorname {BesselY}\left (0,2 \sqrt {e^t}\right )-\operatorname {BesselY}(1,2) \operatorname {BesselJ}\left (0,2 \sqrt {e^t}\right )} \]

✗ Sympy

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

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