8.4.1 problem 1

Internal problem ID [2519]
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 : 1
Date solved : Tuesday, September 30, 2025 at 05:41:47 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=2 t \left (y+1\right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 10

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

\[ y = -1+{\mathrm e}^{t^{2}} \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 12

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

\begin{align*} y(t)&\to e^{t^2}-1 \end{align*}

✓ Sympy. Time used: 0.152 (sec). Leaf size: 8

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

\[ y{\left (t \right )} = e^{t^{2}} - 1 \]