4.17.29 \(-2 x^3 y(x)^2 y'(x)-4 x^2 y(x)^3+y'(x)^2=0\)

ODE
\[ -2 x^3 y(x)^2 y'(x)-4 x^2 y(x)^3+y'(x)^2=0 \] ODE Classification

[[_1st_order, _with_linear_symmetries]]

Book solution method
Change of variable

Mathematica
cpu = 0.587685 (sec), leaf count = 136

\[\left \{\text {Solve}\left [4 c_1+\frac {2 x \sqrt {x^4 y(x)+4} y(x)^{3/2} \sinh ^{-1}\left (\frac {1}{2} x^2 \sqrt {y(x)}\right )}{\sqrt {x^2 y(x)^3 \left (x^4 y(x)+4\right )}}+\log (y(x))=0,y(x)\right ],\text {Solve}\left [4 c_1+\log (y(x))=\frac {2 x y(x)^{3/2} \sqrt {x^4 y(x)+4} \sinh ^{-1}\left (\frac {1}{2} x^2 \sqrt {y(x)}\right )}{\sqrt {x^2 y(x)^3 \left (x^4 y(x)+4\right )}},y(x)\right ]\right \}\]

Maple
cpu = 0.294 (sec), leaf count = 86

\[ \left \{ {x}^{4}+{\frac {{x}^{2}}{y \relax (x ) }\sqrt {{x}^{4} \left (y \relax (x ) \right ) ^{2}+4\,y \relax (x ) }}+2\, \left (y \relax (x ) \right ) ^{-1}-{\it \_C1}=0,{x}^{4} \left (y \relax (x ) \right ) ^{2}+y \relax (x ) {x}^{2}\sqrt {{x}^{4} \left (y \relax (x ) \right ) ^{2}+4\,y \relax (x ) }+2\,y \relax (x ) -{\it \_C1}=0,y \relax (x ) =-4\,{x}^{-4} \right \} \] Mathematica raw input

DSolve[-4*x^2*y[x]^3 - 2*x^3*y[x]^2*y'[x] + y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[4*C[1] + Log[y[x]] + (2*x*ArcSinh[(x^2*Sqrt[y[x]])/2]*y[x]^(3/2)*Sqrt[4 +
 x^4*y[x]])/Sqrt[x^2*y[x]^3*(4 + x^4*y[x])] == 0, y[x]], Solve[4*C[1] + Log[y[x]
] == (2*x*ArcSinh[(x^2*Sqrt[y[x]])/2]*y[x]^(3/2)*Sqrt[4 + x^4*y[x]])/Sqrt[x^2*y[
x]^3*(4 + x^4*y[x])], y[x]]}

Maple raw input

dsolve(diff(y(x),x)^2-2*x^3*y(x)^2*diff(y(x),x)-4*x^2*y(x)^3 = 0, y(x),'implicit')

Maple raw output

y(x) = -4/x^4, x^4*y(x)^2+y(x)*x^2*(x^4*y(x)^2+4*y(x))^(1/2)+2*y(x)-_C1 = 0, x^4
+1/y(x)*x^2*(x^4*y(x)^2+4*y(x))^(1/2)+2/y(x)-_C1 = 0