29.13.25 problem 379

Internal problem ID [4979]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 13
Problem number : 379
Date solved : Monday, January 27, 2025 at 10:00:52 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*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 32

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

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

✓ Solution by Mathematica

Time used: 0.356 (sec). Leaf size: 58

DSolve[x(1-x^4)D[y[x],x]==2 x(x^2-y[x]^2)+(1-x^4) y[x],y[x],x,IncludeSingularSolutions -> True]
 

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