78.4.13 problem 14

Internal problem ID [18136]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Section 8 (Exact Equations). Problems at page 72
Problem number : 14
Date solved : Tuesday, January 28, 2025 at 11:31:37 AM
CAS classification : [_exact, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 30

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

\[ \frac {\left (2 x^{2}-2 y\right ) \sqrt {x^{2}-y}}{3}+x^{2}+c_1 = 0 \]

✓ Solution by Mathematica

Time used: 0.836 (sec). Leaf size: 121

DSolve[2*x*(1+Sqrt[x^2-y[x]])==Sqrt[x^2-y[x]]*D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to x^2+\left (\frac {3}{2}\right )^{2/3} \sqrt [3]{-\left (x^2+c_1\right ){}^2} \\ y(x)\to x^2-\frac {\sqrt [6]{3} \left (\sqrt {3}-3 i\right ) \sqrt [3]{-\left (x^2+c_1\right ){}^2}}{2\ 2^{2/3}} \\ y(x)\to x^2-\frac {\sqrt [6]{3} \left (\sqrt {3}+3 i\right ) \sqrt [3]{-\left (x^2+c_1\right ){}^2}}{2\ 2^{2/3}} \\ \end{align*}