4.19.3 \(-x^4+x^2 y'(x)^2+\left (1-x^2\right ) y(x)^2-2 x y(x) y'(x)=0\)

ODE
\[ -x^4+x^2 y'(x)^2+\left (1-x^2\right ) y(x)^2-2 x y(x) y'(x)=0 \] ODE Classification

[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Book solution method
Change of variable

Mathematica
cpu = 0.0323304 (sec), leaf count = 26

\[\left \{\left \{y(x)\to -x \sinh \left (x-c_1\right )\right \},\left \{y(x)\to x \sinh \left (c_1+x\right )\right \}\right \}\]

Maple
cpu = 1.386 (sec), leaf count = 67

\[ \left \{ {x}^{2}+ \left (y \relax (x ) \right ) ^{2}=0,\ln \left (x \left (y \relax (x ) +\sqrt {{x}^{2}+ \left (y \relax (x ) \right ) ^{2}} \right ) \right ) -x-2\,\ln \relax (x ) -{\it \_C1}=0,\ln \left (x \left (y \relax (x ) +\sqrt {{x}^{2}+ \left (y \relax (x ) \right ) ^{2}} \right ) \right ) +x-2\,\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[-x^4 + (1 - x^2)*y[x]^2 - 2*x*y[x]*y'[x] + x^2*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> -(x*Sinh[x - C[1]])}, {y[x] -> x*Sinh[x + C[1]]}}

Maple raw input

dsolve(x^2*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)-x^4+(-x^2+1)*y(x)^2 = 0, y(x),'implicit')

Maple raw output

x^2+y(x)^2 = 0, ln(x*(y(x)+(x^2+y(x)^2)^(1/2)))+x-2*ln(x)-_C1 = 0, ln(x*(y(x)+(x
^2+y(x)^2)^(1/2)))-x-2*ln(x)-_C1 = 0