60.1.453 problem 456

Internal problem ID [10467]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 456
Date solved : Monday, January 27, 2025 at 07:49:17 PM
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 y^{2}-x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.266 (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*y(x)^2-x = 0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.563 (sec). Leaf size: 111

DSolve[-x + x*y[x]^2 + 2*(1 - x^2)*y[x]*D[y[x],x] + x*(-1 + x^2)*D[y[x],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*}