##### 4.13.10 $$\left (3 x^2-y(x)^2\right ) y'(x)=2 x y(x)$$

ODE
$\left (3 x^2-y(x)^2\right ) y'(x)=2 x y(x)$ ODE Classiﬁcation

[[_homogeneous, class A], _rational, _dAlembert]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.453814 (sec), leaf count = 458

$\left \{\left \{y(x)\to \frac {1}{3} \left (\frac {\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} e^{2 c_1}}{\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-e^{c_1}\right )\right \},\left \{y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{6 \sqrt [3]{2}}-\frac {i \left (\sqrt {3}-i\right ) e^{2 c_1}}{3\ 2^{2/3} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-\frac {e^{c_1}}{3}\right \},\left \{y(x)\to -\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{6 \sqrt [3]{2}}+\frac {i \left (\sqrt {3}+i\right ) e^{2 c_1}}{3\ 2^{2/3} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-\frac {e^{c_1}}{3}\right \}\right \}$

Maple
cpu = 0.061 (sec), leaf count = 402

$\left [y \left (x \right ) = \frac {\left (12 \sqrt {3}\, x \sqrt {27 \textit {\_C1}^{2} x^{2}-4}\, \textit {\_C1} -108 \textit {\_C1}^{2} x^{2}+8\right )^{\frac {1}{3}}}{6 \textit {\_C1}}+\frac {2}{3 \textit {\_C1} \left (12 \sqrt {3}\, x \sqrt {27 \textit {\_C1}^{2} x^{2}-4}\, \textit {\_C1} -108 \textit {\_C1}^{2} x^{2}+8\right )^{\frac {1}{3}}}+\frac {1}{3 \textit {\_C1}}, y \left (x \right ) = -\frac {\left (12 \sqrt {3}\, x \sqrt {27 \textit {\_C1}^{2} x^{2}-4}\, \textit {\_C1} -108 \textit {\_C1}^{2} x^{2}+8\right )^{\frac {1}{3}}}{12 \textit {\_C1}}-\frac {1}{3 \textit {\_C1} \left (12 \sqrt {3}\, x \sqrt {27 \textit {\_C1}^{2} x^{2}-4}\, \textit {\_C1} -108 \textit {\_C1}^{2} x^{2}+8\right )^{\frac {1}{3}}}+\frac {1}{3 \textit {\_C1}}-\frac {i \sqrt {3}\, \left (\frac {\left (12 \sqrt {3}\, x \sqrt {27 \textit {\_C1}^{2} x^{2}-4}\, \textit {\_C1} -108 \textit {\_C1}^{2} x^{2}+8\right )^{\frac {1}{3}}}{6 \textit {\_C1}}-\frac {2}{3 \textit {\_C1} \left (12 \sqrt {3}\, x \sqrt {27 \textit {\_C1}^{2} x^{2}-4}\, \textit {\_C1} -108 \textit {\_C1}^{2} x^{2}+8\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (12 \sqrt {3}\, x \sqrt {27 \textit {\_C1}^{2} x^{2}-4}\, \textit {\_C1} -108 \textit {\_C1}^{2} x^{2}+8\right )^{\frac {1}{3}}}{12 \textit {\_C1}}-\frac {1}{3 \textit {\_C1} \left (12 \sqrt {3}\, x \sqrt {27 \textit {\_C1}^{2} x^{2}-4}\, \textit {\_C1} -108 \textit {\_C1}^{2} x^{2}+8\right )^{\frac {1}{3}}}+\frac {1}{3 \textit {\_C1}}+\frac {i \sqrt {3}\, \left (\frac {\left (12 \sqrt {3}\, x \sqrt {27 \textit {\_C1}^{2} x^{2}-4}\, \textit {\_C1} -108 \textit {\_C1}^{2} x^{2}+8\right )^{\frac {1}{3}}}{6 \textit {\_C1}}-\frac {2}{3 \textit {\_C1} \left (12 \sqrt {3}\, x \sqrt {27 \textit {\_C1}^{2} x^{2}-4}\, \textit {\_C1} -108 \textit {\_C1}^{2} x^{2}+8\right )^{\frac {1}{3}}}\right )}{2}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (-E^C[1] + (2^(1/3)*E^(2*C[1]))/(-2*E^(3*C[1]) + 27*E^C[1]*x^2 + 3*Sqr
t[-12*E^(4*C[1])*x^2 + 81*E^(2*C[1])*x^4])^(1/3) + (-2*E^(3*C[1]) + 27*E^C[1]*x^
2 + 3*Sqrt[-12*E^(4*C[1])*x^2 + 81*E^(2*C[1])*x^4])^(1/3)/2^(1/3))/3}, {y[x] ->
-1/3*E^C[1] - ((I/3)*(-I + Sqrt[3])*E^(2*C[1]))/(2^(2/3)*(-2*E^(3*C[1]) + 27*E^C
[1]*x^2 + 3*Sqrt[-12*E^(4*C[1])*x^2 + 81*E^(2*C[1])*x^4])^(1/3)) + ((I/6)*(I + S
qrt[3])*(-2*E^(3*C[1]) + 27*E^C[1]*x^2 + 3*Sqrt[-12*E^(4*C[1])*x^2 + 81*E^(2*C[1
])*x^4])^(1/3))/2^(1/3)}, {y[x] -> -1/3*E^C[1] + ((I/3)*(I + Sqrt[3])*E^(2*C[1])
)/(2^(2/3)*(-2*E^(3*C[1]) + 27*E^C[1]*x^2 + 3*Sqrt[-12*E^(4*C[1])*x^2 + 81*E^(2*
C[1])*x^4])^(1/3)) - ((I/6)*(-I + Sqrt[3])*(-2*E^(3*C[1]) + 27*E^C[1]*x^2 + 3*Sq
rt[-12*E^(4*C[1])*x^2 + 81*E^(2*C[1])*x^4])^(1/3))/2^(1/3)}}

Maple raw input

dsolve((3*x^2-y(x)^2)*diff(y(x),x) = 2*x*y(x), y(x))

Maple raw output

[y(x) = 1/6/_C1*(12*3^(1/2)*x*(27*_C1^2*x^2-4)^(1/2)*_C1-108*_C1^2*x^2+8)^(1/3)+
2/3/_C1/(12*3^(1/2)*x*(27*_C1^2*x^2-4)^(1/2)*_C1-108*_C1^2*x^2+8)^(1/3)+1/3/_C1,
 y(x) = -1/12/_C1*(12*3^(1/2)*x*(27*_C1^2*x^2-4)^(1/2)*_C1-108*_C1^2*x^2+8)^(1/3
)-1/3/_C1/(12*3^(1/2)*x*(27*_C1^2*x^2-4)^(1/2)*_C1-108*_C1^2*x^2+8)^(1/3)+1/3/_C
1-1/2*I*3^(1/2)*(1/6/_C1*(12*3^(1/2)*x*(27*_C1^2*x^2-4)^(1/2)*_C1-108*_C1^2*x^2+
8)^(1/3)-2/3/_C1/(12*3^(1/2)*x*(27*_C1^2*x^2-4)^(1/2)*_C1-108*_C1^2*x^2+8)^(1/3)
), y(x) = -1/12/_C1*(12*3^(1/2)*x*(27*_C1^2*x^2-4)^(1/2)*_C1-108*_C1^2*x^2+8)^(1
/3)-1/3/_C1/(12*3^(1/2)*x*(27*_C1^2*x^2-4)^(1/2)*_C1-108*_C1^2*x^2+8)^(1/3)+1/3/
_C1+1/2*I*3^(1/2)*(1/6/_C1*(12*3^(1/2)*x*(27*_C1^2*x^2-4)^(1/2)*_C1-108*_C1^2*x^
2+8)^(1/3)-2/3/_C1/(12*3^(1/2)*x*(27*_C1^2*x^2-4)^(1/2)*_C1-108*_C1^2*x^2+8)^(1/
3))]