15.8.37 problem 39

Internal problem ID [3040]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 12, page 46
Problem number : 39
Date solved : Monday, January 27, 2025 at 07:11:10 AM
CAS classification : [_separable]

\begin{align*} x \sqrt {1-y}-\sqrt {-x^{2}+1}\, y^{\prime }&=0 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.198 (sec). Leaf size: 22

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

\[ y = \frac {x^{2}}{4}-\frac {\sqrt {-x^{2}+1}}{2}+\frac {1}{2} \]

✓ Solution by Mathematica

Time used: 0.250 (sec). Leaf size: 53

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

\begin{align*} y(x)\to \frac {1}{4} \left (x^2-2 \sqrt {1-x^2}+2\right ) \\ y(x)\to \frac {1}{4} \left (x^2+6 \sqrt {1-x^2}-6\right ) \\ \end{align*}