ODE
\[ y''(x)=a+4 b^2 y(x)+3 b y(x)^2+3 y(x) y'(x) \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 8.91458 (sec), leaf count = 2742
\[\left \{\left \{y(x)\to \frac {-i^{\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}} 2^{\frac {3 \left (b^3+\sqrt {4 b^6-3 a b^3}\right )}{2 b^3}} 3^{\frac {\sqrt {4 b^3-3 a}}{2 b^{3/2}}} a^{\frac {\sqrt {4 b^3-3 a}}{2 b^{3/2}}} b^{\frac {\sqrt {4 b^6-3 a b^3}}{b^3}+3} \left (\frac {e^{-2 b x}}{a}\right )^{\frac {\sqrt {4 b^3-3 a}}{2 b^{3/2}}} I_{\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {a} \sqrt {\frac {e^{-2 b x}}{a}} \sqrt {c_1}}{b}\right ) \Gamma \left (\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}+1\right ) c_1{}^{\frac {\sqrt {4 b^3-3 a}}{2 b^{3/2}}}+i^{\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}} 2^{\frac {3 \sqrt {4 b^6-3 a b^3}}{2 b^3}+\frac {1}{2}} 3^{\frac {\sqrt {4 b^3-3 a}}{2 b^{3/2}}} a^{\frac {\sqrt {4 b^3-3 a}}{2 b^{3/2}}} b^{\frac {\sqrt {4 b^6-3 a b^3}}{b^3}+\frac {3}{2}} \sqrt {4 b^3-3 a} \left (\frac {e^{-2 b x}}{a}\right )^{\frac {\sqrt {4 b^3-3 a}}{2 b^{3/2}}} I_{\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {a} \sqrt {\frac {e^{-2 b x}}{a}} \sqrt {c_1}}{b}\right ) \Gamma \left (\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}+1\right ) c_1{}^{\frac {\sqrt {4 b^3-3 a}}{2 b^{3/2}}}-i^{\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}} 2^{\frac {3 \sqrt {4 b^6-3 a b^3}}{2 b^3}+\frac {1}{2}} 3^{\frac {\sqrt {4 b^3-3 a}}{2 b^{3/2}}} a^{\frac {\sqrt {4 b^3-3 a}}{2 b^{3/2}}} b^{\frac {\sqrt {4 b^6-3 a b^3}}{b^3}} \sqrt {4 b^6-3 a b^3} \left (\frac {e^{-2 b x}}{a}\right )^{\frac {\sqrt {4 b^3-3 a}}{2 b^{3/2}}} I_{\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {a} \sqrt {\frac {e^{-2 b x}}{a}} \sqrt {c_1}}{b}\right ) \Gamma \left (\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}+1\right ) c_1{}^{\frac {\sqrt {4 b^3-3 a}}{2 b^{3/2}}}+2^{\frac {3 \sqrt {4 b^3-3 a}}{2 b^{3/2}}+\frac {1}{2}} 3^{\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}} a^{\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}} b^{\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}} \left (-2 b^3-\sqrt {4 b^3-3 a} b^{3/2}+\sqrt {4 b^6-3 a b^3}\right ) \left (\frac {e^{-2 b x}}{a}\right )^{\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}} I_{-\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {a} \sqrt {\frac {e^{-2 b x}}{a}} \sqrt {c_1}}{b}\right ) c_2 \Gamma \left (1-\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}\right ) c_1{}^{\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}}+i^{\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}} 2^{\frac {3 \sqrt {4 b^6-3 a b^3}}{2 b^3}} 3^{\frac {1}{2} \left (\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}+1\right )} a^{\frac {1}{2} \left (\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}+1\right )} b^{\frac {\sqrt {4 b^6-3 a b^3}}{b^3}+2} \left (\frac {e^{-2 b x}}{a}\right )^{\frac {1}{2} \left (\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}+1\right )} I_{\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}-1}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {a} \sqrt {\frac {e^{-2 b x}}{a}} \sqrt {c_1}}{b}\right ) \Gamma \left (\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}+1\right ) c_1{}^{\frac {1}{2} \left (\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}+1\right )}+i^{\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}} 2^{\frac {3 \sqrt {4 b^6-3 a b^3}}{2 b^3}} 3^{\frac {1}{2} \left (\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}+1\right )} a^{\frac {1}{2} \left (\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}+1\right )} b^{\frac {\sqrt {4 b^6-3 a b^3}}{b^3}+2} \left (\frac {e^{-2 b x}}{a}\right )^{\frac {1}{2} \left (\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}+1\right )} I_{\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}+1}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {a} \sqrt {\frac {e^{-2 b x}}{a}} \sqrt {c_1}}{b}\right ) \Gamma \left (\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}+1\right ) c_1{}^{\frac {1}{2} \left (\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}+1\right )}+2^{\frac {3 \sqrt {4 b^3-3 a}}{2 b^{3/2}}} 3^{\frac {b^3+\sqrt {4 b^6-3 a b^3}}{2 b^3}} a^{\frac {b^3+\sqrt {4 b^6-3 a b^3}}{2 b^3}} b^{\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}+2} \left (\frac {e^{-2 b x}}{a}\right )^{\frac {b^3+\sqrt {4 b^6-3 a b^3}}{2 b^3}} I_{-\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}-1}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {a} \sqrt {\frac {e^{-2 b x}}{a}} \sqrt {c_1}}{b}\right ) c_2 \Gamma \left (1-\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}\right ) c_1{}^{\frac {b^3+\sqrt {4 b^6-3 a b^3}}{2 b^3}}+2^{\frac {3 \sqrt {4 b^3-3 a}}{2 b^{3/2}}} 3^{\frac {b^3+\sqrt {4 b^6-3 a b^3}}{2 b^3}} a^{\frac {b^3+\sqrt {4 b^6-3 a b^3}}{2 b^3}} b^{\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}+2} \left (\frac {e^{-2 b x}}{a}\right )^{\frac {b^3+\sqrt {4 b^6-3 a b^3}}{2 b^3}} I_{1-\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {a} \sqrt {\frac {e^{-2 b x}}{a}} \sqrt {c_1}}{b}\right ) c_2 \Gamma \left (1-\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}\right ) c_1{}^{\frac {b^3+\sqrt {4 b^6-3 a b^3}}{2 b^3}}}{3 \sqrt {2} b^2 \left (i^{\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}} 2^{\frac {3 \sqrt {4 b^6-3 a b^3}}{2 b^3}} 3^{\frac {\sqrt {4 b^3-3 a}}{2 b^{3/2}}} a^{\frac {\sqrt {4 b^3-3 a}}{2 b^{3/2}}} b^{\frac {\sqrt {4 b^6-3 a b^3}}{b^3}} \left (\frac {e^{-2 b x}}{a}\right )^{\frac {\sqrt {4 b^3-3 a}}{2 b^{3/2}}} I_{\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {a} \sqrt {\frac {e^{-2 b x}}{a}} \sqrt {c_1}}{b}\right ) \Gamma \left (\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}+1\right ) c_1{}^{\frac {\sqrt {4 b^3-3 a}}{2 b^{3/2}}}+2^{\frac {3 \sqrt {4 b^3-3 a}}{2 b^{3/2}}} 3^{\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}} a^{\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}} b^{\frac {\sqrt {4 b^3-3 a}}{b^{3/2}}} \left (\frac {e^{-2 b x}}{a}\right )^{\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}} I_{-\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {a} \sqrt {\frac {e^{-2 b x}}{a}} \sqrt {c_1}}{b}\right ) c_2 \Gamma \left (1-\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}\right ) c_1{}^{\frac {\sqrt {4 b^6-3 a b^3}}{2 b^3}}\right )}\right \}\right \}\]
Maple ✓
cpu = 1.539 (sec), leaf count = 889
\[\left [\int _{}^{y \left (x \right )}-\frac {2 b^{2}}{-4 \textit {\_a} \,b^{3}-3 \textit {\_a}^{2} b^{2}+\RootOf \left (6 \BesselK \left (\frac {-4 b^{3}+3 a}{2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b}, -\frac {\sqrt {3}\, \textit {\_Z}}{2 b^{2}}\right ) \textit {\_C1} \,b^{2}+9 \BesselK \left (\frac {-4 b^{3}+3 a}{2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b}, -\frac {\sqrt {3}\, \textit {\_Z}}{2 b^{2}}\right ) \textit {\_C1} b \textit {\_a} -3 \BesselK \left (\frac {-4 b^{3}+3 a}{2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b}, -\frac {\sqrt {3}\, \textit {\_Z}}{2 b^{2}}\right ) \sqrt {4 b^{4}-3 a b}\, \textit {\_C1} +3 \BesselK \left (\frac {4 b^{3}+2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b -3 a}{2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b}, -\frac {\sqrt {3}\, \textit {\_Z}}{2 b^{2}}\right ) \sqrt {3}\, \textit {\_C1} \textit {\_Z} +6 \BesselI \left (-\frac {-4 b^{3}+3 a}{2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b}, -\frac {\sqrt {3}\, \textit {\_Z}}{2 b^{2}}\right ) b^{2}+9 \BesselI \left (-\frac {-4 b^{3}+3 a}{2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b}, -\frac {\sqrt {3}\, \textit {\_Z}}{2 b^{2}}\right ) b \textit {\_a} -3 \BesselI \left (\frac {4 b^{3}+2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b -3 a}{2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b}, -\frac {\sqrt {3}\, \textit {\_Z}}{2 b^{2}}\right ) \sqrt {3}\, \textit {\_Z} -3 \BesselI \left (-\frac {-4 b^{3}+3 a}{2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b}, -\frac {\sqrt {3}\, \textit {\_Z}}{2 b^{2}}\right ) \sqrt {4 b^{4}-3 a b}\right )^{2}-a b}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {2 b^{2}}{-4 \textit {\_a} \,b^{3}-3 \textit {\_a}^{2} b^{2}+\RootOf \left (6 \BesselK \left (\frac {-4 b^{3}+3 a}{2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b}, \frac {\sqrt {3}\, \textit {\_Z}}{2 b^{2}}\right ) \textit {\_C1} \,b^{2}+9 \BesselK \left (\frac {-4 b^{3}+3 a}{2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b}, \frac {\sqrt {3}\, \textit {\_Z}}{2 b^{2}}\right ) \textit {\_C1} b \textit {\_a} -3 \BesselK \left (\frac {-4 b^{3}+3 a}{2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b}, \frac {\sqrt {3}\, \textit {\_Z}}{2 b^{2}}\right ) \sqrt {4 b^{4}-3 a b}\, \textit {\_C1} +3 \BesselK \left (\frac {4 b^{3}+2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b -3 a}{2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b}, \frac {\sqrt {3}\, \textit {\_Z}}{2 b^{2}}\right ) \sqrt {3}\, \textit {\_C1} \textit {\_Z} +6 \BesselI \left (-\frac {-4 b^{3}+3 a}{2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b}, \frac {\sqrt {3}\, \textit {\_Z}}{2 b^{2}}\right ) b^{2}+9 \BesselI \left (-\frac {-4 b^{3}+3 a}{2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b}, \frac {\sqrt {3}\, \textit {\_Z}}{2 b^{2}}\right ) b \textit {\_a} -3 \BesselI \left (\frac {4 b^{3}+2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b -3 a}{2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b}, \frac {\sqrt {3}\, \textit {\_Z}}{2 b^{2}}\right ) \sqrt {3}\, \textit {\_Z} -3 \BesselI \left (-\frac {-4 b^{3}+3 a}{2 \sqrt {-\left (-4 b^{3}+3 a \right ) b}\, b}, \frac {\sqrt {3}\, \textit {\_Z}}{2 b^{2}}\right ) \sqrt {4 b^{4}-3 a b}\right )^{2}-a b}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] Mathematica raw input
DSolve[y''[x] == a + 4*b^2*y[x] + 3*b*y[x]^2 + 3*y[x]*y'[x],y[x],x]
Mathematica raw output
{{y[x] -> (2^(1/2 + (3*Sqrt[-3*a + 4*b^3])/(2*b^(3/2)))*3^(Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3))*a^(Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3))*b^(Sqrt[-3*a + 4*b^3]/b^(3/2))*(-2*b^3 - b^(3/2)*Sqrt[-3*a + 4*b^3] + Sqrt[-3*a*b^3 + 4*b^6])*(1/(a*E^(2*b*x)))^(Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3))*BesselI[-1/2*Sqrt[-3*a*b^3 + 4*b^6]/b^3, (Sqrt[3/2]*Sqrt[a]*Sqrt[1/(a*E^(2*b*x))]*Sqrt[C[1]])/b]*C[1]^(Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3))*C[2]*Gamma[1 - Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3)] + 2^((3*Sqrt[-3*a + 4*b^3])/(2*b^(3/2)))*3^((b^3 + Sqrt[-3*a*b^3 + 4*b^6])/(2*b^3))*a^((b^3 + Sqrt[-3*a*b^3 + 4*b^6])/(2*b^3))*b^(2 + Sqrt[-3*a + 4*b^3]/b^(3/2))*(1/(a*E^(2*b*x)))^((b^3 + Sqrt[-3*a*b^3 + 4*b^6])/(2*b^3))*BesselI[-1 - Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3), (Sqrt[3/2]*Sqrt[a]*Sqrt[1/(a*E^(2*b*x))]*Sqrt[C[1]])/b]*C[1]^((b^3 + Sqrt[-3*a*b^3 + 4*b^6])/(2*b^3))*C[2]*Gamma[1 - Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3)] + 2^((3*Sqrt[-3*a + 4*b^3])/(2*b^(3/2)))*3^((b^3 + Sqrt[-3*a*b^3 + 4*b^6])/(2*b^3))*a^((b^3 + Sqrt[-3*a*b^3 + 4*b^6])/(2*b^3))*b^(2 + Sqrt[-3*a + 4*b^3]/b^(3/2))*(1/(a*E^(2*b*x)))^((b^3 + Sqrt[-3*a*b^3 + 4*b^6])/(2*b^3))*BesselI[1 - Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3), (Sqrt[3/2]*Sqrt[a]*Sqrt[1/(a*E^(2*b*x))]*Sqrt[C[1]])/b]*C[1]^((b^3 + Sqrt[-3*a*b^3 + 4*b^6])/(2*b^3))*C[2]*Gamma[1 - Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3)] - I^(Sqrt[-3*a + 4*b^3]/b^(3/2))*2^((3*(b^3 + Sqrt[-3*a*b^3 + 4*b^6]))/(2*b^3))*3^(Sqrt[-3*a + 4*b^3]/(2*b^(3/2)))*a^(Sqrt[-3*a + 4*b^3]/(2*b^(3/2)))*b^(3 + Sqrt[-3*a*b^3 + 4*b^6]/b^3)*(1/(a*E^(2*b*x)))^(Sqrt[-3*a + 4*b^3]/(2*b^(3/2)))*BesselI[Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3), (Sqrt[3/2]*Sqrt[a]*Sqrt[1/(a*E^(2*b*x))]*Sqrt[C[1]])/b]*C[1]^(Sqrt[-3*a + 4*b^3]/(2*b^(3/2)))*Gamma[1 + Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3)] + I^(Sqrt[-3*a + 4*b^3]/b^(3/2))*2^(1/2 + (3*Sqrt[-3*a*b^3 + 4*b^6])/(2*b^3))*3^(Sqrt[-3*a + 4*b^3]/(2*b^(3/2)))*a^(Sqrt[-3*a + 4*b^3]/(2*b^(3/2)))*b^(3/2 + Sqrt[-3*a*b^3 + 4*b^6]/b^3)*Sqrt[-3*a + 4*b^3]*(1/(a*E^(2*b*x)))^(Sqrt[-3*a + 4*b^3]/(2*b^(3/2)))*BesselI[Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3), (Sqrt[3/2]*Sqrt[a]*Sqrt[1/(a*E^(2*b*x))]*Sqrt[C[1]])/b]*C[1]^(Sqrt[-3*a + 4*b^3]/(2*b^(3/2)))*Gamma[1 + Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3)] - I^(Sqrt[-3*a + 4*b^3]/b^(3/2))*2^(1/2 + (3*Sqrt[-3*a*b^3 + 4*b^6])/(2*b^3))*3^(Sqrt[-3*a + 4*b^3]/(2*b^(3/2)))*a^(Sqrt[-3*a + 4*b^3]/(2*b^(3/2)))*b^(Sqrt[-3*a*b^3 + 4*b^6]/b^3)*Sqrt[-3*a*b^3 + 4*b^6]*(1/(a*E^(2*b*x)))^(Sqrt[-3*a + 4*b^3]/(2*b^(3/2)))*BesselI[Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3), (Sqrt[3/2]*Sqrt[a]*Sqrt[1/(a*E^(2*b*x))]*Sqrt[C[1]])/b]*C[1]^(Sqrt[-3*a + 4*b^3]/(2*b^(3/2)))*Gamma[1 + Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3)] + I^(Sqrt[-3*a + 4*b^3]/b^(3/2))*2^((3*Sqrt[-3*a*b^3 + 4*b^6])/(2*b^3))*3^((1 + Sqrt[-3*a + 4*b^3]/b^(3/2))/2)*a^((1 + Sqrt[-3*a + 4*b^3]/b^(3/2))/2)*b^(2 + Sqrt[-3*a*b^3 + 4*b^6]/b^3)*(1/(a*E^(2*b*x)))^((1 + Sqrt[-3*a + 4*b^3]/b^(3/2))/2)*BesselI[-1 + Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3), (Sqrt[3/2]*Sqrt[a]*Sqrt[1/(a*E^(2*b*x))]*Sqrt[C[1]])/b]*C[1]^((1 + Sqrt[-3*a + 4*b^3]/b^(3/2))/2)*Gamma[1 + Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3)] + I^(Sqrt[-3*a + 4*b^3]/b^(3/2))*2^((3*Sqrt[-3*a*b^3 + 4*b^6])/(2*b^3))*3^((1 + Sqrt[-3*a + 4*b^3]/b^(3/2))/2)*a^((1 + Sqrt[-3*a + 4*b^3]/b^(3/2))/2)*b^(2 + Sqrt[-3*a*b^3 + 4*b^6]/b^3)*(1/(a*E^(2*b*x)))^((1 + Sqrt[-3*a + 4*b^3]/b^(3/2))/2)*BesselI[1 + Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3), (Sqrt[3/2]*Sqrt[a]*Sqrt[1/(a*E^(2*b*x))]*Sqrt[C[1]])/b]*C[1]^((1 + Sqrt[-3*a + 4*b^3]/b^(3/2))/2)*Gamma[1 + Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3)])/(3*Sqrt[2]*b^2*(2^((3*Sqrt[-3*a + 4*b^3])/(2*b^(3/2)))*3^(Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3))*a^(Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3))*b^(Sqrt[-3*a + 4*b^3]/b^(3/2))*(1/(a*E^(2*b*x)))^(Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3))*BesselI[-1/2*Sqrt[-3*a*b^3 + 4*b^6]/b^3, (Sqrt[3/2]*Sqrt[a]*Sqrt[1/(a*E^(2*b*x))]*Sqrt[C[1]])/b]*C[1]^(Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3))*C[2]*Gamma[1 - Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3)] + I^(Sqrt[-3*a + 4*b^3]/b^(3/2))*2^((3*Sqrt[-3*a*b^3 + 4*b^6])/(2*b^3))*3^(Sqrt[-3*a + 4*b^3]/(2*b^(3/2)))*a^(Sqrt[-3*a + 4*b^3]/(2*b^(3/2)))*b^(Sqrt[-3*a*b^3 + 4*b^6]/b^3)*(1/(a*E^(2*b*x)))^(Sqrt[-3*a + 4*b^3]/(2*b^(3/2)))*BesselI[Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3), (Sqrt[3/2]*Sqrt[a]*Sqrt[1/(a*E^(2*b*x))]*Sqrt[C[1]])/b]*C[1]^(Sqrt[-3*a + 4*b^3]/(2*b^(3/2)))*Gamma[1 + Sqrt[-3*a*b^3 + 4*b^6]/(2*b^3)]))}}
Maple raw input
dsolve(diff(diff(y(x),x),x) = 3*y(x)*diff(y(x),x)+a+4*b^2*y(x)+3*b*y(x)^2, y(x))
Maple raw output
[Intat(-2/(-4*_a*b^3-3*_a^2*b^2+RootOf(6*BesselK(1/2*(-4*b^3+3*a)/(-(-4*b^3+3*a)*b)^(1/2)/b,-1/2/b^2*3^(1/2)*_Z)*_C1*b^2+9*BesselK(1/2*(-4*b^3+3*a)/(-(-4*b^3+3*a)*b)^(1/2)/b,-1/2/b^2*3^(1/2)*_Z)*_C1*b*_a-3*BesselK(1/2*(-4*b^3+3*a)/(-(-4*b^3+3*a)*b)^(1/2)/b,-1/2/b^2*3^(1/2)*_Z)*(4*b^4-3*a*b)^(1/2)*_C1+3*BesselK(1/2*(4*b^3+2*(-(-4*b^3+3*a)*b)^(1/2)*b-3*a)/(-(-4*b^3+3*a)*b)^(1/2)/b,-1/2/b^2*3^(1/2)*_Z)*3^(1/2)*_C1*_Z+6*BesselI(-1/2*(-4*b^3+3*a)/(-(-4*b^3+3*a)*b)^(1/2)/b,-1/2/b^2*3^(1/2)*_Z)*b^2+9*BesselI(-1/2*(-4*b^3+3*a)/(-(-4*b^3+3*a)*b)^(1/2)/b,-1/2/b^2*3^(1/2)*_Z)*b*_a-3*BesselI(1/2*(4*b^3+2*(-(-4*b^3+3*a)*b)^(1/2)*b-3*a)/(-(-4*b^3+3*a)*b)^(1/2)/b,-1/2/b^2*3^(1/2)*_Z)*3^(1/2)*_Z-3*BesselI(-1/2*(-4*b^3+3*a)/(-(-4*b^3+3*a)*b)^(1/2)/b,-1/2/b^2*3^(1/2)*_Z)*(4*b^4-3*a*b)^(1/2))^2-a*b)*b^2,_a = y(x))-x-_C2 = 0, Intat(-2/(-4*_a*b^3-3*_a^2*b^2+RootOf(6*BesselK(1/2*(-4*b^3+3*a)/(-(-4*b^3+3*a)*b)^(1/2)/b,1/2/b^2*3^(1/2)*_Z)*_C1*b^2+9*BesselK(1/2*(-4*b^3+3*a)/(-(-4*b^3+3*a)*b)^(1/2)/b,1/2/b^2*3^(1/2)*_Z)*_C1*b*_a-3*BesselK(1/2*(-4*b^3+3*a)/(-(-4*b^3+3*a)*b)^(1/2)/b,1/2/b^2*3^(1/2)*_Z)*(4*b^4-3*a*b)^(1/2)*_C1+3*BesselK(1/2*(4*b^3+2*(-(-4*b^3+3*a)*b)^(1/2)*b-3*a)/(-(-4*b^3+3*a)*b)^(1/2)/b,1/2/b^2*3^(1/2)*_Z)*3^(1/2)*_C1*_Z+6*BesselI(-1/2*(-4*b^3+3*a)/(-(-4*b^3+3*a)*b)^(1/2)/b,1/2/b^2*3^(1/2)*_Z)*b^2+9*BesselI(-1/2*(-4*b^3+3*a)/(-(-4*b^3+3*a)*b)^(1/2)/b,1/2/b^2*3^(1/2)*_Z)*b*_a-3*BesselI(1/2*(4*b^3+2*(-(-4*b^3+3*a)*b)^(1/2)*b-3*a)/(-(-4*b^3+3*a)*b)^(1/2)/b,1/2/b^2*3^(1/2)*_Z)*3^(1/2)*_Z-3*BesselI(-1/2*(-4*b^3+3*a)/(-(-4*b^3+3*a)*b)^(1/2)/b,1/2/b^2*3^(1/2)*_Z)*(4*b^4-3*a*b)^(1/2))^2-a*b)*b^2,_a = y(x))-x-_C2 = 0]