4.2.36 \(y(x)^3 \left (a+4 b^2 x+3 b x^2\right )+y'(x)+3 x y(x)^2=0\)

ODE
\[ y(x)^3 \left (a+4 b^2 x+3 b x^2\right )+y'(x)+3 x y(x)^2=0 \] ODE Classification

[_Abel]

Book solution method
Abel ODE, First kind

Mathematica
cpu = 9.14762 (sec), leaf count = 376

\[\text {Solve}\left [c_1=\frac {\left (3 x-b \left (\sqrt {4-\frac {3 a}{b^3}}-2\right )\right ) J_{\frac {1}{2} \sqrt {4-\frac {3 a}{b^3}}}\left (-\frac {1}{2} i \sqrt {3} \sqrt {\frac {(a+b x (4 b+3 x)) y(x)-2 b}{b^3 y(x)}}\right )-i \sqrt {3} b \sqrt {\frac {y(x) (a+b x (4 b+3 x))-2 b}{b^3 y(x)}} J_{\frac {1}{2} \left (\sqrt {4-\frac {3 a}{b^3}}+2\right )}\left (-\frac {1}{2} i \sqrt {3} \sqrt {\frac {(a+b x (4 b+3 x)) y(x)-2 b}{b^3 y(x)}}\right )}{i \sqrt {3} b \sqrt {\frac {y(x) (a+b x (4 b+3 x))-2 b}{b^3 y(x)}} Y_{\frac {1}{2} \left (\sqrt {4-\frac {3 a}{b^3}}+2\right )}\left (-\frac {1}{2} i \sqrt {3} \sqrt {\frac {(a+b x (4 b+3 x)) y(x)-2 b}{b^3 y(x)}}\right )+\left (b \left (\sqrt {4-\frac {3 a}{b^3}}-2\right )-3 x\right ) Y_{\frac {1}{2} \sqrt {4-\frac {3 a}{b^3}}}\left (-\frac {1}{2} i \sqrt {3} \sqrt {\frac {(a+b x (4 b+3 x)) y(x)-2 b}{b^3 y(x)}}\right )},y(x)\right ]\]

Maple
cpu = 1.876 (sec), leaf count = 373

\[ \left \{ {\it \_C1}+{1 \left (-{{\sl K}_{{\frac {1}{2}\sqrt {{\frac {4\,{b}^{3}-3\,a}{{b}^{3}}}}}+1}\left (-{\frac {\sqrt {3}}{2}\sqrt {{\frac {4\,{b}^{2}xy \relax (x ) + \left (3\,{x}^{2}y \relax (x ) -2 \right ) b+ay \relax (x ) }{{b}^{3}y \relax (x ) }}}}\right )}\sqrt {3}\sqrt {{\frac {4\,{b}^{2}xy \relax (x ) + \left (3\,{x}^{2}y \relax (x ) -2 \right ) b+ay \relax (x ) }{{b}^{3}y \relax (x ) }}}b-{{\sl K}_{{\frac {1}{2}\sqrt {{\frac {4\,{b}^{3}-3\,a}{{b}^{3}}}}}}\left (-{\frac {\sqrt {3}}{2}\sqrt {{\frac {4\,{b}^{2}xy \relax (x ) + \left (3\,{x}^{2}y \relax (x ) -2 \right ) b+ay \relax (x ) }{{b}^{3}y \relax (x ) }}}}\right )} \left (b\sqrt {{\frac {4\,{b}^{3}-3\,a}{{b}^{3}}}}-2\,b-3\,x \right ) \right ) \left ({{\sl I}_{{\frac {1}{2}\sqrt {{\frac {4\,{b}^{3}-3\,a}{{b}^{3}}}}}+1}\left (-{\frac {\sqrt {3}}{2}\sqrt {{\frac {4\,{b}^{2}xy \relax (x ) + \left (3\,{x}^{2}y \relax (x ) -2 \right ) b+ay \relax (x ) }{{b}^{3}y \relax (x ) }}}}\right )}\sqrt {3}\sqrt {{\frac {4\,{b}^{2}xy \relax (x ) + \left (3\,{x}^{2}y \relax (x ) -2 \right ) b+ay \relax (x ) }{{b}^{3}y \relax (x ) }}}b-{{\sl I}_{{\frac {1}{2}\sqrt {{\frac {4\,{b}^{3}-3\,a}{{b}^{3}}}}}}\left (-{\frac {\sqrt {3}}{2}\sqrt {{\frac {4\,{b}^{2}xy \relax (x ) + \left (3\,{x}^{2}y \relax (x ) -2 \right ) b+ay \relax (x ) }{{b}^{3}y \relax (x ) }}}}\right )} \left (b\sqrt {{\frac {4\,{b}^{3}-3\,a}{{b}^{3}}}}-2\,b-3\,x \right ) \right ) ^{-1}}=0 \right \} \] Mathematica raw input

DSolve[3*x*y[x]^2 + (a + 4*b^2*x + 3*b*x^2)*y[x]^3 + y'[x] == 0,y[x],x]

Mathematica raw output

Solve[C[1] == ((-((-2 + Sqrt[4 - (3*a)/b^3])*b) + 3*x)*BesselJ[Sqrt[4 - (3*a)/b^
3]/2, (-I/2)*Sqrt[3]*Sqrt[(-2*b + (a + b*x*(4*b + 3*x))*y[x])/(b^3*y[x])]] - I*S
qrt[3]*b*BesselJ[(2 + Sqrt[4 - (3*a)/b^3])/2, (-I/2)*Sqrt[3]*Sqrt[(-2*b + (a + b
*x*(4*b + 3*x))*y[x])/(b^3*y[x])]]*Sqrt[(-2*b + (a + b*x*(4*b + 3*x))*y[x])/(b^3
*y[x])])/(((-2 + Sqrt[4 - (3*a)/b^3])*b - 3*x)*BesselY[Sqrt[4 - (3*a)/b^3]/2, (-
I/2)*Sqrt[3]*Sqrt[(-2*b + (a + b*x*(4*b + 3*x))*y[x])/(b^3*y[x])]] + I*Sqrt[3]*b
*BesselY[(2 + Sqrt[4 - (3*a)/b^3])/2, (-I/2)*Sqrt[3]*Sqrt[(-2*b + (a + b*x*(4*b 
+ 3*x))*y[x])/(b^3*y[x])]]*Sqrt[(-2*b + (a + b*x*(4*b + 3*x))*y[x])/(b^3*y[x])])
, y[x]]

Maple raw input

dsolve(diff(y(x),x)+3*x*y(x)^2+(4*b^2*x+3*b*x^2+a)*y(x)^3 = 0, y(x),'implicit')

Maple raw output

_C1+(-BesselK(1/2*((4*b^3-3*a)/b^3)^(1/2)+1,-1/2*3^(1/2)*((4*b^2*x*y(x)+(3*x^2*y
(x)-2)*b+a*y(x))/b^3/y(x))^(1/2))*3^(1/2)*((4*b^2*x*y(x)+(3*x^2*y(x)-2)*b+a*y(x)
)/b^3/y(x))^(1/2)*b-BesselK(1/2*((4*b^3-3*a)/b^3)^(1/2),-1/2*3^(1/2)*((4*b^2*x*y
(x)+(3*x^2*y(x)-2)*b+a*y(x))/b^3/y(x))^(1/2))*(b*((4*b^3-3*a)/b^3)^(1/2)-2*b-3*x
))/(BesselI(1/2*((4*b^3-3*a)/b^3)^(1/2)+1,-1/2*3^(1/2)*((4*b^2*x*y(x)+(3*x^2*y(x
)-2)*b+a*y(x))/b^3/y(x))^(1/2))*3^(1/2)*((4*b^2*x*y(x)+(3*x^2*y(x)-2)*b+a*y(x))/
b^3/y(x))^(1/2)*b-BesselI(1/2*((4*b^3-3*a)/b^3)^(1/2),-1/2*3^(1/2)*((4*b^2*x*y(x
)+(3*x^2*y(x)-2)*b+a*y(x))/b^3/y(x))^(1/2))*(b*((4*b^3-3*a)/b^3)^(1/2)-2*b-3*x))
 = 0