83.3.5 problem 5

Internal problem ID [18982]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Exercise II (B) at page 9
Problem number : 5
Date solved : Thursday, March 13, 2025 at 01:16:33 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (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 \left (x \right ) &= \frac {\sqrt {\left (x^{2}+1\right ) \left (-x^{2}+c_{1} \right )}}{x^{2}+1} \\ y \left (x \right ) &= -\frac {\sqrt {\left (x^{2}+1\right ) \left (-x^{2}+c_{1} \right )}}{x^{2}+1} \\ \end{align*}

✓ Mathematica. Time used: 0.362 (sec). Leaf size: 129

ode=(x*y[x]^2+x)+(y[x]*x^2+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 {-x^2-1}}{\sqrt {x^2+1}} \\ y(x)\to \frac {\sqrt {-x^2-1}}{\sqrt {x^2+1}} \\ \end{align*}

✓ Sympy. Time used: 0.681 (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)
 

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