76.1.28 problem 28

Internal problem ID [17256]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.1 (Separable equations). Problems at page 44
Problem number : 28
Date solved : Monday, March 31, 2025 at 03:47:25 PM
CAS classification : [_separable]

\begin{align*} y^{2} \sqrt {-x^{2}+1}\, y^{\prime }&=\arcsin \left (x \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.154 (sec). Leaf size: 16

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

\[ y = \frac {\left (8+12 \arcsin \left (x \right )^{2}\right )^{{1}/{3}}}{2} \]

✓ Mathematica. Time used: 0.538 (sec). Leaf size: 19

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

\[ y(x)\to \sqrt [3]{\frac {3 \arcsin (x)^2}{2}+1} \]

✓ Sympy. Time used: 1.420 (sec). Leaf size: 15

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

\[ y{\left (x \right )} = \sqrt [3]{\frac {3 \operatorname {asin}^{2}{\left (x \right )}}{2} + 1} \]