14.4.19 problem 19

Internal problem ID [2537]
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 : 19
Date solved : Sunday, March 30, 2025 at 12:09:21 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=t \sqrt {1-y^{2}} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 5

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

\[ y = 1 \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 6

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

\[ y(t)\to 1 \]

✓ Sympy. Time used: 0.297 (sec). Leaf size: 8

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

\[ y{\left (t \right )} = \cos {\left (\frac {t^{2}}{2} \right )} \]