38.2.46 problem 46

Internal problem ID [6475]
Book : Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY. 2001
Section : Program 24. First order differential equations. Further problems 24. page 1068
Problem number : 46
Date solved : Wednesday, March 05, 2025 at 12:51:27 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 29

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

\begin{align*} y &= \sqrt {c_1 \,x^{2}+c_1 -1} \\ y &= -\sqrt {c_1 \,x^{2}+c_1 -1} \\ \end{align*}

✓ Mathematica. Time used: 0.53 (sec). Leaf size: 61

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

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

✓ Sympy. Time used: 0.607 (sec). Leaf size: 29

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

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