29.31.30 problem 931

Internal problem ID [5509]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 31
Problem number : 931
Date solved : Monday, January 27, 2025 at 11:34:30 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Solution by Maple

Time used: 0.289 (sec). Leaf size: 33

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

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

✓ Solution by Mathematica

Time used: 2.680 (sec). Leaf size: 111

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

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