7.5.15 problem 19

Internal problem ID [2361]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 1.10. Page 80
Problem number : 19
Date solved : Tuesday, September 30, 2025 at 05:34:47 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,op(ic)],y(t), singsol=all);
 

\[ y = 1 \]

✓ Mathematica. Time used: 0.001 (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]
 

\begin{align*} y(t)&\to 1 \end{align*}

✓ Sympy. Time used: 0.176 (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)
 

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