4.14 problem 14

Internal problem ID [5442]

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.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [_exact, [_1st_order, _with_symmetry_[F(x),G(y)]]]

Solve \begin {gather*} \boxed {2 x \left (1+\sqrt {x^{2}-y}\right )-\sqrt {x^{2}-y}\, y^{\prime }=0} \end {gather*}

Solution by Maple

Time used: 0.016 (sec). Leaf size: 20

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

\[ x^{2}+\frac {2 \left (x^{2}-y \relax (x )\right )^{\frac {3}{2}}}{3}+c_{1} = 0 \]

Solution by Mathematica

Time used: 1.423 (sec). Leaf size: 121

DSolve[2*x*(1+Sqrt[x^2-y[x]])==Sqrt[x^2-y[x]]*y'[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*}