29.18.27 problem 505

Internal problem ID [5101]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 18
Problem number : 505
Date solved : Sunday, March 30, 2025 at 06:40:20 AM
CAS classification : [_separable]

\begin{align*} x y y^{\prime }+1+y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 34

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

\begin{align*} y &= \frac {\sqrt {-x^{2}+c_1}}{x} \\ y &= -\frac {\sqrt {-x^{2}+c_1}}{x} \\ \end{align*}

✓ Mathematica. Time used: 0.344 (sec). Leaf size: 96

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

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

✓ Sympy. Time used: 0.513 (sec). Leaf size: 26

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

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