4.12.7 $x(3y(x)+2x)y^{\prime }(x)=y(x{)}^2$

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

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

Book solution method
Homogeneous equation

Mathematica ✓
cpu = 0.0881313 (sec), leaf count = 413

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

Maple ✓
cpu = 0.019 (sec), leaf count = 31

$$\{-\frac{1}{2}\ln \left(\frac{x+y\left(x\right)}{x}\right)-\ln \left(\frac{y\left(x\right)}{x}\right)-\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (-x + x^2/((27*E^C[1]*x)/2 - x^3 + (3*Sqrt[3]*Sqrt[E^C[1]*x^2*(27*E^C[
1] - 4*x^2)])/2)^(1/3) + ((27*E^C[1]*x)/2 - x^3 + (3*Sqrt[3]*Sqrt[E^C[1]*x^2*(27
*E^C[1] - 4*x^2)])/2)^(1/3))/3}, {y[x] -> (-4*x - (2*(1 + I*Sqrt[3])*x^2)/((27*E
^C[1]*x)/2 - x^3 + (3*Sqrt[3]*Sqrt[E^C[1]*x^2*(27*E^C[1] - 4*x^2)])/2)^(1/3) + I
*2^(2/3)*(I + Sqrt[3])*(27*E^C[1]*x - 2*x^3 + 3*Sqrt[3]*Sqrt[E^C[1]*x^2*(27*E^C[
1] - 4*x^2)])^(1/3))/12}, {y[x] -> (-4*x + ((2*I)*(I + Sqrt[3])*x^2)/((27*E^C[1]
*x)/2 - x^3 + (3*Sqrt[3]*Sqrt[E^C[1]*x^2*(27*E^C[1] - 4*x^2)])/2)^(1/3) - 2^(2/3
)*(1 + I*Sqrt[3])*(27*E^C[1]*x - 2*x^3 + 3*Sqrt[3]*Sqrt[E^C[1]*x^2*(27*E^C[1] - 
4*x^2)])^(1/3))/12}}

Maple raw input

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

Maple raw output

-1/2*ln((x+y(x))/x)-ln(y(x)/x)-ln(x)-_C1 = 0