82.11.4 problem Ex. 4

Internal problem ID [18685]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Exercises at page 28
Problem number : Ex. 4
Date solved : Thursday, March 13, 2025 at 12:37:42 PM
CAS classification : [_rational, _Bernoulli]

\begin{align*} y^{\prime }+\frac {x y}{-x^{2}+1}&=x \sqrt {y} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 27

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

\[ \sqrt {y \left (x \right )}+\frac {1}{3}-\frac {x^{2}}{3}-\left (x -1\right )^{{1}/{4}} \left (x +1\right )^{{1}/{4}} c_{1} = 0 \]

✓ Mathematica. Time used: 0.226 (sec). Leaf size: 29

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

\[ y(x)\to \frac {1}{9} \left (x^2+3 c_1 \sqrt [4]{x^2-1}-1\right ){}^2 \]

✓ Sympy. Time used: 0.495 (sec). Leaf size: 60

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

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