ODE
\[ a+x^3 y'(x)^2+x^2 y(x) y'(x)=0 \] ODE Classification
[[_homogeneous, `class G`]]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 0.367889 (sec), leaf count = 123
\[\left \{\left \{y(x)\to -\frac {e^{-\frac {c_1}{2}} \left (2 a e^{c_1}+x\right )}{\sqrt {2} x}\right \},\left \{y(x)\to \frac {e^{-\frac {c_1}{2}} \left (2 a e^{c_1}+x\right )}{\sqrt {2} x}\right \},\left \{y(x)\to -\frac {e^{-\frac {c_1}{2}} \left (2 a x+e^{c_1}\right )}{\sqrt {2} x}\right \},\left \{y(x)\to \frac {e^{-\frac {c_1}{2}} \left (2 a x+e^{c_1}\right )}{\sqrt {2} x}\right \}\right \}\]
Maple ✓
cpu = 0.138 (sec), leaf count = 66
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{2}-4\,{\frac {a}{x}}=0,xy \left ( x \right ) -{\it \_C1}+\sqrt {{x}^{2} \left ( y \left ( x \right ) \right ) ^{2}-4\,ax}=0,y \left ( x \right ) +{\frac {1}{x}\sqrt {{x}^{2} \left ( y \left ( x \right ) \right ) ^{2}-4\,ax}}-{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[a + x^2*y[x]*y'[x] + x^3*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -((2*a*E^C[1] + x)/(Sqrt[2]*E^(C[1]/2)*x))}, {y[x] -> (2*a*E^C[1] + x)/(Sqrt[2]*E^(C[1]/2)*x)}, {y[x] -> -((E^C[1] + 2*a*x)/(Sqrt[2]*E^(C[1]/2)*x))}, {y[x] -> (E^C[1] + 2*a*x)/(Sqrt[2]*E^(C[1]/2)*x)}}
Maple raw input
dsolve(x^3*diff(y(x),x)^2+x^2*y(x)*diff(y(x),x)+a = 0, y(x),'implicit')
Maple raw output
y(x)^2-4/x*a = 0, y(x)+1/x*(x^2*y(x)^2-4*a*x)^(1/2)-_C1 = 0, x*y(x)-_C1+(x^2*y(x)^2-4*a*x)^(1/2) = 0