50.4.14 problem 14

Internal problem ID [7863]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.5. Exact Equations. Page 20
Problem number : 14
Date solved : Monday, January 27, 2025 at 03:28:00 PM
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: 0.004 (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: 0.860 (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*}