ODE
\[ -x^2-2 x y(x) y'(x)+3 y(x)^2 y'(x)^2+4 y(x)^2=0 \] ODE Classification
[[_homogeneous, `class A`], _dAlembert]
Book solution method
Homogeneous ODE, \(x^n f\left ( \frac {y}{x} , y' \right )=0\), Solve for \(x\)
Mathematica ✓
cpu = 0.329332 (sec), leaf count = 76
\[\left \{\left \{y(x)\to -\frac {\sqrt {-3 x^2-4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}}\right \},\left \{y(x)\to \frac {\sqrt {-3 x^2-4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}}\right \}\right \}\]
Maple ✓
cpu = 1.867 (sec), leaf count = 203
\[\left [y \left (x \right ) = -\frac {\sqrt {3}\, x}{3}, y \left (x \right ) = \frac {\sqrt {3}\, x}{3}, \ln \left (x \right )-\frac {\sqrt {\frac {x^{2}-3 y \left (x \right )^{2}}{x^{2}}}}{2}+\arctanh \left (\frac {\sqrt {\frac {x^{2}-3 y \left (x \right )^{2}}{x^{2}}}}{2}\right )+\frac {\sqrt {3}\, \sqrt {\frac {\left (\sqrt {3}\, x +3 y \left (x \right )\right ) \left (\sqrt {3}\, x -3 y \left (x \right )\right )}{x^{2}}}}{6}+\frac {\ln \left (\frac {x^{2}+y \left (x \right )^{2}}{x^{2}}\right )}{2}-\textit {\_C1} = 0, \ln \left (x \right )+\frac {\sqrt {\frac {x^{2}-3 y \left (x \right )^{2}}{x^{2}}}}{2}-\arctanh \left (\frac {\sqrt {\frac {x^{2}-3 y \left (x \right )^{2}}{x^{2}}}}{2}\right )-\frac {\sqrt {3}\, \sqrt {\frac {\left (\sqrt {3}\, x +3 y \left (x \right )\right ) \left (\sqrt {3}\, x -3 y \left (x \right )\right )}{x^{2}}}}{6}+\frac {\ln \left (\frac {x^{2}+y \left (x \right )^{2}}{x^{2}}\right )}{2}-\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[-x^2 + 4*y[x]^2 - 2*x*y[x]*y'[x] + 3*y[x]^2*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -(Sqrt[E^(6*C[1]) - (4*I)*E^(3*C[1])*x - 3*x^2]/Sqrt[3])}, {y[x] -> Sqrt[E^(6*C[1]) - (4*I)*E^(3*C[1])*x - 3*x^2]/Sqrt[3]}}
Maple raw input
dsolve(3*y(x)^2*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)-x^2+4*y(x)^2 = 0, y(x))
Maple raw output
[y(x) = -1/3*3^(1/2)*x, y(x) = 1/3*3^(1/2)*x, ln(x)-1/2*((x^2-3*y(x)^2)/x^2)^(1/2)+arctanh(1/2*((x^2-3*y(x)^2)/x^2)^(1/2))+1/6*3^(1/2)*((3^(1/2)*x+3*y(x))*(3^(1/2)*x-3*y(x))/x^2)^(1/2)+1/2*ln((x^2+y(x)^2)/x^2)-_C1 = 0, ln(x)+1/2*((x^2-3*y(x)^2)/x^2)^(1/2)-arctanh(1/2*((x^2-3*y(x)^2)/x^2)^(1/2))-1/6*3^(1/2)*((3^(1/2)*x+3*y(x))*(3^(1/2)*x-3*y(x))/x^2)^(1/2)+1/2*ln((x^2+y(x)^2)/x^2)-_C1 = 0]