29.24.7 problem 669

Internal problem ID [5260]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 24
Problem number : 669
Date solved : Sunday, March 30, 2025 at 07:33:09 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 40

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

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (\ln \left (x^{2}+1\right ) {\mathrm e}^{\textit {\_Z}}+2 c_1 \,{\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-\ln \left (x^{2}+1\right )-2 c_1 -\textit {\_Z} -2\right )} \]

✓ Mathematica. Time used: 0.346 (sec). Leaf size: 40

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

\begin{align*} y(x)\to \text {InverseFunction}\left [\log (\text {$\#$1})-\frac {2}{\text {$\#$1}-1}\&\right ]\left [-\log \left (x^2+1\right )+c_1\right ] \\ y(x)\to 0 \\ y(x)\to 1 \\ \end{align*}

✓ Sympy. Time used: 0.407 (sec). Leaf size: 19

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

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