15.1.6 problem 6

Internal problem ID [2846]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 5, page 21
Problem number : 6
Date solved : Sunday, March 30, 2025 at 12:33:59 AM
CAS classification : [_separable]

\begin{align*} x y^{2}+x +\left (x^{2} y-y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 54

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

\begin{align*} y &= \frac {\sqrt {\left (x^{2}-1\right ) \left (-x^{2}+c_1 \right )}}{x^{2}-1} \\ y &= -\frac {\sqrt {\left (x^{2}-1\right ) \left (-x^{2}+c_1 \right )}}{x^{2}-1} \\ \end{align*}

✓ Mathematica. Time used: 0.389 (sec). Leaf size: 133

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

\begin{align*} y(x)\to -\frac {\sqrt {-x^2+1-e^{2 c_1}}}{\sqrt {x^2-1}} \\ y(x)\to \frac {\sqrt {-x^2+1-e^{2 c_1}}}{\sqrt {x^2-1}} \\ y(x)\to -i \\ y(x)\to i \\ y(x)\to -\frac {\sqrt {1-x^2}}{\sqrt {x^2-1}} \\ y(x)\to \frac {\sqrt {1-x^2}}{\sqrt {x^2-1}} \\ \end{align*}

✓ Sympy. Time used: 0.898 (sec). Leaf size: 32

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

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