4.12.7 \(x (3 y(x)+2 x) y'(x)=y(x)^2\)

ODE
\[ x (3 y(x)+2 x) y'(x)=y(x)^2 \] ODE Classification

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

Book solution method
Homogeneous equation

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

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

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

\[ \left \{ -{\frac {1}{2}\ln \left ({\frac {x+y \relax (x ) }{x}} \right ) }-\ln \left ({\frac {y \relax (x ) }{x}} \right ) -\ln \relax (x ) -{\it \_C1}=0 \right \} \] 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