71.8.39 problem 13 (c)

Internal problem ID [14473]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.4.4, page 115
Problem number : 13 (c)
Date solved : Tuesday, January 28, 2025 at 06:41:23 AM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&={\frac {1}{2}} \end{align*}

✓ Solution by Maple

Time used: 0.049 (sec). Leaf size: 14

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

\[ y = \sin \left (\frac {x^{2}}{2}+\frac {\pi }{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.116 (sec). Leaf size: 17

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

\[ y(x)\to \sin \left (\frac {1}{6} \left (3 x^2+\pi \right )\right ) \]