4.13.25 \((y(x)+3 x)^2 y'(x)=4 y(x) (2 y(x)+3 x)\)

ODE
\[ (y(x)+3 x)^2 y'(x)=4 y(x) (2 y(x)+3 x) \] ODE Classification

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0719511 (sec), leaf count = 747

\[\left \{\left \{y(x)\to \frac {1}{4} \left (-\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (e^{c_1}-16 x\right )}-16 e^{c_1} x+e^{2 c_1}+16 x^2}-\sqrt {2} \sqrt {-6 \sqrt [3]{-e^{c_1} x^4 \left (e^{c_1}-16 x\right )}+\frac {\left (e^{c_1}-8 x\right ){}^3-72 x^2 \left (e^{c_1}-8 x\right )}{\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (e^{c_1}-16 x\right )}-16 e^{c_1} x+e^{2 c_1}+16 x^2}}+\left (e^{c_1}-8 x\right ){}^2-48 x^2}-e^{c_1}+8 x\right )\right \},\left \{y(x)\to \frac {1}{4} \left (-\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (e^{c_1}-16 x\right )}-16 e^{c_1} x+e^{2 c_1}+16 x^2}+\sqrt {2} \sqrt {-6 \sqrt [3]{-e^{c_1} x^4 \left (e^{c_1}-16 x\right )}+\frac {\left (e^{c_1}-8 x\right ){}^3-72 x^2 \left (e^{c_1}-8 x\right )}{\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (e^{c_1}-16 x\right )}-16 e^{c_1} x+e^{2 c_1}+16 x^2}}+\left (e^{c_1}-8 x\right ){}^2-48 x^2}-e^{c_1}+8 x\right )\right \},\left \{y(x)\to \frac {1}{4} \left (\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (e^{c_1}-16 x\right )}-16 e^{c_1} x+e^{2 c_1}+16 x^2}-\sqrt {2} \sqrt {-6 \sqrt [3]{-e^{c_1} x^4 \left (e^{c_1}-16 x\right )}+\frac {72 x^2 \left (e^{c_1}-8 x\right )-\left (e^{c_1}-8 x\right ){}^3}{\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (e^{c_1}-16 x\right )}-16 e^{c_1} x+e^{2 c_1}+16 x^2}}+\left (e^{c_1}-8 x\right ){}^2-48 x^2}-e^{c_1}+8 x\right )\right \},\left \{y(x)\to \frac {1}{4} \left (\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (e^{c_1}-16 x\right )}-16 e^{c_1} x+e^{2 c_1}+16 x^2}+\sqrt {2} \sqrt {-6 \sqrt [3]{-e^{c_1} x^4 \left (e^{c_1}-16 x\right )}+\frac {72 x^2 \left (e^{c_1}-8 x\right )-\left (e^{c_1}-8 x\right ){}^3}{\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (e^{c_1}-16 x\right )}-16 e^{c_1} x+e^{2 c_1}+16 x^2}}+\left (e^{c_1}-8 x\right ){}^2-48 x^2}-e^{c_1}+8 x\right )\right \}\right \}\]

Maple
cpu = 0.027 (sec), leaf count = 44

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

DSolve[(3*x + y[x])^2*y'[x] == 4*y[x]*(3*x + 2*y[x]),y[x],x]

Mathematica raw output

{{y[x] -> (-E^C[1] + 8*x - Sqrt[E^(2*C[1]) - 16*E^C[1]*x + 16*x^2 + 12*(-(E^C[1]
*(E^C[1] - 16*x)*x^4))^(1/3)] - Sqrt[2]*Sqrt[(E^C[1] - 8*x)^2 - 48*x^2 - 6*(-(E^
C[1]*(E^C[1] - 16*x)*x^4))^(1/3) + ((E^C[1] - 8*x)^3 - 72*(E^C[1] - 8*x)*x^2)/Sq
rt[E^(2*C[1]) - 16*E^C[1]*x + 16*x^2 + 12*(-(E^C[1]*(E^C[1] - 16*x)*x^4))^(1/3)]
])/4}, {y[x] -> (-E^C[1] + 8*x - Sqrt[E^(2*C[1]) - 16*E^C[1]*x + 16*x^2 + 12*(-(
E^C[1]*(E^C[1] - 16*x)*x^4))^(1/3)] + Sqrt[2]*Sqrt[(E^C[1] - 8*x)^2 - 48*x^2 - 6
*(-(E^C[1]*(E^C[1] - 16*x)*x^4))^(1/3) + ((E^C[1] - 8*x)^3 - 72*(E^C[1] - 8*x)*x
^2)/Sqrt[E^(2*C[1]) - 16*E^C[1]*x + 16*x^2 + 12*(-(E^C[1]*(E^C[1] - 16*x)*x^4))^
(1/3)]])/4}, {y[x] -> (-E^C[1] + 8*x + Sqrt[E^(2*C[1]) - 16*E^C[1]*x + 16*x^2 + 
12*(-(E^C[1]*(E^C[1] - 16*x)*x^4))^(1/3)] - Sqrt[2]*Sqrt[(E^C[1] - 8*x)^2 - 48*x
^2 - 6*(-(E^C[1]*(E^C[1] - 16*x)*x^4))^(1/3) + (-(E^C[1] - 8*x)^3 + 72*(E^C[1] -
 8*x)*x^2)/Sqrt[E^(2*C[1]) - 16*E^C[1]*x + 16*x^2 + 12*(-(E^C[1]*(E^C[1] - 16*x)
*x^4))^(1/3)]])/4}, {y[x] -> (-E^C[1] + 8*x + Sqrt[E^(2*C[1]) - 16*E^C[1]*x + 16
*x^2 + 12*(-(E^C[1]*(E^C[1] - 16*x)*x^4))^(1/3)] + Sqrt[2]*Sqrt[(E^C[1] - 8*x)^2
 - 48*x^2 - 6*(-(E^C[1]*(E^C[1] - 16*x)*x^4))^(1/3) + (-(E^C[1] - 8*x)^3 + 72*(E
^C[1] - 8*x)*x^2)/Sqrt[E^(2*C[1]) - 16*E^C[1]*x + 16*x^2 + 12*(-(E^C[1]*(E^C[1] 
- 16*x)*x^4))^(1/3)]])/4}}

Maple raw input

dsolve((3*x+y(x))^2*diff(y(x),x) = 4*(3*x+2*y(x))*y(x), y(x),'implicit')

Maple raw output

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