\[ -4 a^2 y(x)-3 a y(x)^2-b+y''(x)-3 y(x) y'(x)=0 \] ✓ Mathematica : cpu = 6.73247 (sec), leaf count = 3227
DSolve[-b - 4*a^2*y[x] - 3*a*y[x]^2 - 3*y[x]*Derivative[1][y][x] + Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to -\frac {2 \left ((-1)^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}+1} 2^{-\frac {3 \left (a^{3/2} \sqrt {4 a^3-3 b}-2 a^3\right )}{4 a^3}+\frac {3 \sqrt {4 a^6-3 a^3 b}}{4 a^3}+1} 3^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} a^{-\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{2 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1} b^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}-1} \left (\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}\right ) e^{-2 a x} I_{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right ) c_1{}^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} \Gamma \left (\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1\right ) \left (\frac {e^{-2 a x}}{b}\right )^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}-1}+(-1)^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}+1} 2^{-\frac {3 \left (a^{3/2} \sqrt {4 a^3-3 b}-2 a^3\right )}{4 a^3}+\frac {3 \sqrt {4 a^6-3 a^3 b}}{4 a^3}-\frac {3}{2}} 3^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}+\frac {1}{2}} a^{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}-\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{2 a^3}} b^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}-\frac {1}{2}} e^{-2 a x} \left (I_{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}-1}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right )+I_{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right )\right ) c_1{}^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}+\frac {1}{2}} \Gamma \left (\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1\right ) \left (\frac {e^{-2 a x}}{b}\right )^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}-\frac {1}{2}}+c_2 \left ((-1)^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+1} 2^{-\frac {3 \left (-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}\right )}{4 a^3}-\frac {3 \sqrt {4 a^6-3 a^3 b}}{4 a^3}-\frac {3}{2}} 3^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}+\frac {1}{2}} b^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}-\frac {1}{2}} e^{-2 a x} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}-\frac {1}{2}} \left (I_{-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}-1}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right )+I_{1-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right )\right ) c_1{}^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}+\frac {1}{2}} \Gamma \left (1-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}\right ) a^{-\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{2 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}}+(-1)^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+1} 2^{-\frac {3 \left (-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}\right )}{4 a^3}-\frac {3 \sqrt {4 a^6-3 a^3 b}}{4 a^3}+1} 3^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} b^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}-1} \left (\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}\right ) e^{-2 a x} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}-1} I_{-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right ) c_1{}^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} \Gamma \left (1-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}\right ) a^{-\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{2 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1}\right )\right )}{3 \left ((-1)^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}} 2^{-\frac {3 \left (-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}\right )}{4 a^3}-\frac {3 \sqrt {4 a^6-3 a^3 b}}{4 a^3}} 3^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} b^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} I_{-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right ) c_1{}^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} c_2 \Gamma \left (1-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}\right ) a^{-\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{2 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}}+(-1)^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}} 2^{\frac {3 \sqrt {4 a^6-3 a^3 b}}{4 a^3}-\frac {3 \left (a^{3/2} \sqrt {4 a^3-3 b}-2 a^3\right )}{4 a^3}} 3^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} b^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} I_{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right ) c_1{}^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} \Gamma \left (\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1\right ) a^{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}-\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{2 a^3}}\right )}\right \}\right \}\] ✓ Maple : cpu = 1.345 (sec), leaf count = 783
dsolve(diff(diff(y(x),x),x)-3*y(x)*diff(y(x),x)-3*a*y(x)^2-4*a^2*y(x)-b=0,y(x))
\[\int _{}^{y \left (x \right )}-\frac {6 a^{2}}{-12 \textit {\_a} \,a^{3}-9 \textit {\_a}^{2} a^{2}+\RootOf \left (2 \BesselK \left (\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} a^{2}+3 \BesselK \left (\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} a \textit {\_a} +2 \BesselI \left (-\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) a^{2}+3 \BesselI \left (-\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) a \textit {\_a} +\BesselK \left (\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) \sqrt {4 a^{4}-3 a b}\, c_{1}+\textit {\_Z} \BesselK \left (\frac {-4 a^{3}+2 \sqrt {4 a^{4}-3 a b}\, a +3 b}{2 \sqrt {4 a^{4}-3 a b}\, a}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1}+\BesselI \left (-\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) \sqrt {4 a^{4}-3 a b}-\textit {\_Z} \BesselI \left (\frac {-4 a^{3}+2 \sqrt {4 a^{4}-3 a b}\, a +3 b}{2 \sqrt {4 a^{4}-3 a b}\, a}, -\frac {\textit {\_Z}}{2 a^{2}}\right )\right )^{2}-3 a b}d \textit {\_a} -x -c_{2} = 0\]