64.3.9 problem 10

Internal problem ID [13209]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.1 (Exact differential equations and integrating factors). Exercises page 37
Problem number : 10
Date solved : Monday, March 31, 2025 at 07:37:54 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} \frac {2 y^{{3}/{2}}+1}{\sqrt {x}}+\left (3 \sqrt {x}\, \sqrt {y}-1\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 36

ode:=(2*y(x)^(3/2)+1)/x^(1/2)+(3*x^(1/2)*y(x)^(1/2)-1)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ 2 \sqrt {2 y^{{3}/{2}}+1}\, \sqrt {x}-\int _{}^{y}\frac {1}{\sqrt {2 \textit {\_a}^{{3}/{2}}+1}}d \textit {\_a} +c_1 = 0 \]

✓ Mathematica. Time used: 0.19 (sec). Leaf size: 54

ode=(2*y[x]^(3/2)+1)/x^(1/2)+(3*x^(1/2)*y[x]^(1/2)-1)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {8}{3} \sqrt {x} \sqrt {2 y(x)^{3/2}+1}-\frac {4}{3} y(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},-2 y(x)^{3/2}\right )=c_1,y(x)\right ] \]

✓ Sympy. Time used: 3.038 (sec). Leaf size: 60

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((3*sqrt(x)*sqrt(y(x)) - 1)*Derivative(y(x), x) + (2*y(x)**(3/2) + 1)/sqrt(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} + 2 \sqrt {x} \sqrt {2 y^{\frac {3}{2}}{\left (x \right )} + 1} - \frac {2 y{\left (x \right )} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {2 e^{i \pi } y^{\frac {3}{2}}{\left (x \right )}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} = 0 \]