ODE
\[ -a+y(x)^3 \left (-y'(x)\right )+2 x y(x)^2 y'(x)^2=0 \] ODE Classification
[[_homogeneous, `class G`], _rational]
Book solution method
No Missing Variables ODE, Solve for \(x\)
Mathematica ✓
cpu = 0.28601 (sec), leaf count = 133
\[\left \{\left \{y(x)\to -\frac {e^{-\frac {c_1}{4}} \sqrt {e^{c_1}-8 a x}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {e^{-\frac {c_1}{4}} \sqrt {e^{c_1}-8 a x}}{\sqrt {2}}\right \},\left \{y(x)\to -\frac {\sqrt {e^{-\frac {c_1}{2}} \left (8 a x-e^{c_1}\right )}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {e^{-\frac {c_1}{2}} \left (8 a x-e^{c_1}\right )}}{\sqrt {2}}\right \}\right \}\]
Maple ✓
cpu = 0.151 (sec), leaf count = 104
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{4}+8\,ax=0,-\ln \left ( x \right ) -\ln \left ( - \left ( y \left ( x \right ) \right ) ^{2}+\sqrt { \left ( y \left ( x \right ) \right ) ^{4}+8\,ax} \right ) +\ln \left ( \left ( y \left ( x \right ) \right ) ^{2}+\sqrt { \left ( y \left ( x \right ) \right ) ^{4}+8\,ax} \right ) +{\it \_C1}=0,-\ln \left ( x \right ) +\ln \left ( - \left ( y \left ( x \right ) \right ) ^{2}+\sqrt { \left ( y \left ( x \right ) \right ) ^{4}+8\,ax} \right ) -\ln \left ( \left ( y \left ( x \right ) \right ) ^{2}+\sqrt { \left ( y \left ( x \right ) \right ) ^{4}+8\,ax} \right ) +{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[-a - y[x]^3*y'[x] + 2*x*y[x]^2*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -(Sqrt[E^C[1] - 8*a*x]/(Sqrt[2]*E^(C[1]/4)))}, {y[x] -> Sqrt[E^C[1] - 8*a*x]/(Sqrt[2]*E^(C[1]/4))}, {y[x] -> -(Sqrt[(-E^C[1] + 8*a*x)/E^(C[1]/2)]/Sqrt[2])}, {y[x] -> Sqrt[(-E^C[1] + 8*a*x)/E^(C[1]/2)]/Sqrt[2]}}
Maple raw input
dsolve(2*x*y(x)^2*diff(y(x),x)^2-y(x)^3*diff(y(x),x)-a = 0, y(x),'implicit')
Maple raw output
y(x)^4+8*a*x = 0, -ln(x)+ln(-y(x)^2+(y(x)^4+8*a*x)^(1/2))-ln(y(x)^2+(y(x)^4+8*a*x)^(1/2))+_C1 = 0, -ln(x)-ln(-y(x)^2+(y(x)^4+8*a*x)^(1/2))+ln(y(x)^2+(y(x)^4+8*a*x)^(1/2))+_C1 = 0