4.13.10 $(3x^2-y(x{)}^2)y^{\prime }(x)=2xy(x)$

ODE
$$(3x^2-y(x{)}^2)y^{\prime }(x)=2xy(x)$$ ODE Classification

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

Book solution method
Exact equation, integrating factor

Mathematica ✓
cpu = 0.178843 (sec), leaf count = 458

$$\{\{y(x)\to \frac{1}{3}(\frac{\sqrt[3]{27e^{c_1}{x}^2+3\sqrt{81e^{2c_1}{x}^4-12e^{4c_1}{x}^2}-2e^{3c_1}}}{\sqrt[3]{2}}+\frac{\sqrt[3]{2}{e}^{2c_1}}{\sqrt[3]{27e^{c_1}{x}^2+3\sqrt{81e^{2c_1}{x}^4-12e^{4c_1}{x}^2}-2e^{3c_1}}}-e^{c_1})\},\{y(x)\to \frac{i(\sqrt{3}+i)\sqrt[3]{27e^{c_1}{x}^2+3\sqrt{81e^{2c_1}{x}^4-12e^{4c_1}{x}^2}-2e^{3c_1}}}{6\sqrt[3]{2}}-\frac{i(\sqrt{3}-i)e^{2c_1}}{32^{2/3}\sqrt[3]{27e^{c_1}{x}^2+3\sqrt{81e^{2c_1}{x}^4-12e^{4c_1}{x}^2}-2e^{3c_1}}}-\frac{e^{c_1}}{3}\},\{y(x)\to -\frac{i(\sqrt{3}-i)\sqrt[3]{27e^{c_1}{x}^2+3\sqrt{81e^{2c_1}{x}^4-12e^{4c_1}{x}^2}-2e^{3c_1}}}{6\sqrt[3]{2}}+\frac{i(\sqrt{3}+i)e^{2c_1}}{32^{2/3}\sqrt[3]{27e^{c_1}{x}^2+3\sqrt{81e^{2c_1}{x}^4-12e^{4c_1}{x}^2}-2e^{3c_1}}}-\frac{e^{c_1}}{3}\}\}$$

Maple ✓
cpu = 0.025 (sec), leaf count = 40

$$\{\ln \left(\frac{y\left(x\right)-x}{x}\right)+\ln \left(\frac{x+y\left(x\right)}{x}\right)-3\ln \left(\frac{y\left(x\right)}{x}\right)-\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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] -> 
-E^C[1]/3 - ((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 + Sqr
t[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] -> -E^C[1]/3 + ((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*Sqrt[-
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),'implicit')

Maple raw output

ln((y(x)-x)/x)+ln((x+y(x))/x)-3*ln(y(x)/x)-ln(x)-_C1 = 0