76.1.28 problem 28

Internal problem ID [17327]
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 : Tuesday, January 28, 2025 at 10:01:02 AM
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*}

✓ Solution by Maple

Time used: 0.196 (sec). Leaf size: 16

dsolve([y(x)^2*sqrt(1-x^2)*diff(y(x),x)=arcsin(x),y(0) = 1],y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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