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 = 0.473682 (sec), leaf count = 747
\[\left \{\left \{y(x)\to \frac {1}{4} \left (-\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}+16 x^2-16 e^{c_1} x+e^{2 c_1}}-\sqrt {2} \sqrt {-6 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}-48 x^2+\frac {\left (-8 x+e^{c_1}\right ){}^3-72 x^2 \left (-8 x+e^{c_1}\right )}{\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}+16 x^2-16 e^{c_1} x+e^{2 c_1}}}+\left (-8 x+e^{c_1}\right ){}^2}+8 x-e^{c_1}\right )\right \},\left \{y(x)\to \frac {1}{4} \left (-\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}+16 x^2-16 e^{c_1} x+e^{2 c_1}}+\sqrt {2} \sqrt {-6 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}-48 x^2+\frac {\left (-8 x+e^{c_1}\right ){}^3-72 x^2 \left (-8 x+e^{c_1}\right )}{\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}+16 x^2-16 e^{c_1} x+e^{2 c_1}}}+\left (-8 x+e^{c_1}\right ){}^2}+8 x-e^{c_1}\right )\right \},\left \{y(x)\to \frac {1}{4} \left (\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}+16 x^2-16 e^{c_1} x+e^{2 c_1}}-\sqrt {2} \sqrt {-6 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}-48 x^2+\frac {72 x^2 \left (-8 x+e^{c_1}\right )-\left (-8 x+e^{c_1}\right ){}^3}{\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}+16 x^2-16 e^{c_1} x+e^{2 c_1}}}+\left (-8 x+e^{c_1}\right ){}^2}+8 x-e^{c_1}\right )\right \},\left \{y(x)\to \frac {1}{4} \left (\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}+16 x^2-16 e^{c_1} x+e^{2 c_1}}+\sqrt {2} \sqrt {-6 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}-48 x^2+\frac {72 x^2 \left (-8 x+e^{c_1}\right )-\left (-8 x+e^{c_1}\right ){}^3}{\sqrt {12 \sqrt [3]{-e^{c_1} x^4 \left (-16 x+e^{c_1}\right )}+16 x^2-16 e^{c_1} x+e^{2 c_1}}}+\left (-8 x+e^{c_1}\right ){}^2}+8 x-e^{c_1}\right )\right \}\right \}\]
Maple ✓
cpu = 0.128 (sec), leaf count = 47
\[\left [3 \ln \left (\frac {y \left (x \right )}{x}\right )-\ln \left (\frac {x +y \left (x \right )}{x}\right )-3 \ln \left (-\frac {3 x -y \left (x \right )}{x}\right )-\ln \left (x \right )-\textit {\_C1} = 0\right ]\] 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)/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}, {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))
Maple raw output
[3*ln(y(x)/x)-ln((x+y(x))/x)-3*ln(-(3*x-y(x))/x)-ln(x)-_C1 = 0]