29.7.14 problem 189

Internal problem ID [4789]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 7
Problem number : 189
Date solved : Sunday, March 30, 2025 at 03:54:48 AM
CAS classification : [_rational, _Bernoulli]

\begin{align*} x y^{\prime }+y&=a \left (x^{2}+1\right ) y^{3} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 43

ode:=x*diff(y(x),x)+y(x) = a*(x^2+1)*y(x)^3; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.678 (sec). Leaf size: 68

ode=x D[y[x],x]+y[x]==a(1+x^2)y[x]^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {1}{\sqrt {3 a x^2-2 a x^2 \log (x)+a+c_1 x^2}} \\ y(x)\to \frac {1}{\sqrt {3 a x^2-2 a x^2 \log (x)+a+c_1 x^2}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.516 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*(x**2 + 1)*y(x)**3 + x*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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