23.1.397 problem 382

Internal problem ID [5004]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 382
Date solved : Tuesday, September 30, 2025 at 11:20:09 AM
CAS classification : [[_homogeneous, `class D`], _rational, _Riccati]

\begin{align*} x \left (-x^{4}+1\right ) y^{\prime }&=2 x \left (x^{2}-y^{2}\right )+\left (-x^{4}+1\right ) y \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 32

ode:=x*(-x^4+1)*diff(y(x),x) = 2*x*(x^2-y(x)^2)+(-x^4+1)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\tanh \left (\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}-\frac {\ln \left (x^{2}+1\right )}{2}+2 c_1 \right ) x \]

✓ Mathematica. Time used: 0.078 (sec). Leaf size: 63

ode=x*(1-x^4)*D[y[x],x]==2*x*(x^2-y[x]^2)+(1-x^4)*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {1}{(K[1]-1) (K[1]+1)}dK[1]=\int _1^x\frac {2 K[2]}{(K[2]-1) (K[2]+1) \left (K[2]^2+1\right )}dK[2]+c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.260 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(1 - x**4)*Derivative(y(x), x) - 2*x*(x**2 - y(x)**2) - (1 - x**4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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