4.13.29 \(\left (x^2-3 y(x)^2\right ) y'(x)+2 x y(x)+1=0\)

ODE
\[ \left (x^2-3 y(x)^2\right ) y'(x)+2 x y(x)+1=0 \] ODE Classification

[_exact, _rational]

Book solution method
Exact equation

Mathematica
cpu = 0.0162062 (sec), leaf count = 307

\[\left \{\left \{y(x)\to -\frac {\sqrt [3]{\sqrt {729 \left (x-c_1\right ){}^2-108 x^6}+27 c_1-27 x}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{\sqrt {729 \left (x-c_1\right ){}^2-108 x^6}+27 c_1-27 x}}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {729 \left (x-c_1\right ){}^2-108 x^6}+27 c_1-27 x}}{6 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{\sqrt {729 \left (x-c_1\right ){}^2-108 x^6}+27 c_1-27 x}}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {729 \left (x-c_1\right ){}^2-108 x^6}+27 c_1-27 x}}{6 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{\sqrt {729 \left (x-c_1\right ){}^2-108 x^6}+27 c_1-27 x}}\right \}\right \}\]

Maple
cpu = 0.014 (sec), leaf count = 18

\[ \left \{ {x}^{2}y \relax (x ) - \left (y \relax (x ) \right ) ^{3}+{\it \_C1}+x=0 \right \} \] Mathematica raw input

DSolve[1 + 2*x*y[x] + (x^2 - 3*y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -((2^(1/3)*x^2)/(-27*x + Sqrt[-108*x^6 + 729*(x - C[1])^2] + 27*C[1])^
(1/3)) - (-27*x + Sqrt[-108*x^6 + 729*(x - C[1])^2] + 27*C[1])^(1/3)/(3*2^(1/3))
}, {y[x] -> ((1 + I*Sqrt[3])*x^2)/(2^(2/3)*(-27*x + Sqrt[-108*x^6 + 729*(x - C[1
])^2] + 27*C[1])^(1/3)) + ((1 - I*Sqrt[3])*(-27*x + Sqrt[-108*x^6 + 729*(x - C[1
])^2] + 27*C[1])^(1/3))/(6*2^(1/3))}, {y[x] -> ((1 - I*Sqrt[3])*x^2)/(2^(2/3)*(-
27*x + Sqrt[-108*x^6 + 729*(x - C[1])^2] + 27*C[1])^(1/3)) + ((1 + I*Sqrt[3])*(-
27*x + Sqrt[-108*x^6 + 729*(x - C[1])^2] + 27*C[1])^(1/3))/(6*2^(1/3))}}

Maple raw input

dsolve((x^2-3*y(x)^2)*diff(y(x),x)+1+2*x*y(x) = 0, y(x),'implicit')

Maple raw output

x^2*y(x)-y(x)^3+_C1+x = 0