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 : Wednesday, March 05, 2025 at 05:10:58 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*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 30

ode:=2*x*(1+(x^2-y(x))^(1/2)) = (x^2-y(x))^(1/2)*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.86 (sec). Leaf size: 121

ode=2*x*(1+Sqrt[x^2-y[x]])==Sqrt[x^2-y[x]]*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 5.186 (sec). Leaf size: 109

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*(sqrt(x**2 - y(x)) + 1) - sqrt(x**2 - y(x))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = x^{2} + \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{- 9 C_{1}^{2} - 9 C_{1} x^{2} - \frac {9 x^{4}}{4}}}{2}, \ y{\left (x \right )} = x^{2} + \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{- 9 C_{1}^{2} - 9 C_{1} x^{2} - \frac {9 x^{4}}{4}}}{2}, \ y{\left (x \right )} = x^{2} + \sqrt [3]{- 9 C_{1}^{2} - 9 C_{1} x^{2} - \frac {9 x^{4}}{4}}\right ] \]