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^{\prime } y-x^{4}+\left (-x^{2}+1\right ) y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.664 (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 ) c_{1} {\mathrm e}^{-x}}{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 ) = c_{1} x \\ \end{align*}
✓ Solution by Mathematica
Time used: 38.394 (sec). Leaf size: 99
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 -\frac {x \tanh (x-c_1)}{\sqrt {\text {sech}^2(x-c_1)}} \\ y(x)\to \frac {x \tanh (x-c_1)}{\sqrt {\text {sech}^2(x-c_1)}} \\ y(x)\to -\frac {x \tanh (x+c_1)}{\sqrt {\text {sech}^2(x+c_1)}} \\ y(x)\to \frac {x \tanh (x+c_1)}{\sqrt {\text {sech}^2(x+c_1)}} \\ \end{align*}