Internal problem ID [8016]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 436.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {x^{2} \left (y^{\prime }\right )^{2}-2 x y y^{\prime }+y^{2} \left (-x^{2}+1\right )-x^{4}=0} \end {gather*}
✓ Solution by Maple
Time used: 1.828 (sec). Leaf size: 51
dsolve(x^2*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+y(x)^2*(-x^2+1)-x^4 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {x \left (\frac {{\mathrm e}^{2 x}}{c_{1}^{2}}-1\right ) {\mathrm e}^{-x} c_{1}}{2} \\ y \relax (x ) = \frac {x \left ({\mathrm e}^{2 x} c_{1}^{2}-1\right ) {\mathrm e}^{-x}}{2 c_{1}} \\ y \relax (x ) = x c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.216 (sec). Leaf size: 26
DSolve[-x^4 + (1 - x^2)*y[x]^2 - 2*x*y[x]*y'[x] + x^2*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x \sinh (x+c_1) \\ y(x)\to -x \sinh (x-c_1) \\ \end{align*}