##### 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 = 7.21006 (sec), leaf count = 352

$\text {Solve}\left [c_1=\frac {\left (3 x-b \left (\sqrt {4-\frac {3 a}{b^3}}-2\right )\right ) J_{\frac {1}{2} \sqrt {4-\frac {3 a}{b^3}}}\left (-\frac {1}{2} i \sqrt {\frac {3 (a+b x (4 b+3 x)) y(x)-6 b}{b^3 y(x)}}\right )-i b \sqrt {\frac {3 y(x) (a+b x (4 b+3 x))-6 b}{b^3 y(x)}} J_{\frac {1}{2} \left (\sqrt {4-\frac {3 a}{b^3}}+2\right )}\left (-\frac {1}{2} i \sqrt {\frac {3 (a+b x (4 b+3 x)) y(x)-6 b}{b^3 y(x)}}\right )}{i b \sqrt {\frac {3 y(x) (a+b x (4 b+3 x))-6 b}{b^3 y(x)}} Y_{\frac {1}{2} \left (\sqrt {4-\frac {3 a}{b^3}}+2\right )}\left (-\frac {1}{2} i \sqrt {\frac {3 (a+b x (4 b+3 x)) y(x)-6 b}{b^3 y(x)}}\right )+\left (b \left (\sqrt {4-\frac {3 a}{b^3}}-2\right )-3 x\right ) Y_{\frac {1}{2} \sqrt {4-\frac {3 a}{b^3}}}\left (-\frac {1}{2} i \sqrt {\frac {3 (a+b x (4 b+3 x)) y(x)-6 b}{b^3 y(x)}}\right )},y(x)\right ]$

Maple
cpu = 1.459 (sec), leaf count = 373

$\left [\textit {\_C1} +\frac {-\BesselK \left (\frac {\sqrt {\frac {4 b^{3}-3 a}{b^{3}}}}{2}+1, -\frac {\sqrt {3}\, \sqrt {\frac {4 b^{2} x y \left (x \right )+\left (3 x^{2} y \left (x \right )-2\right ) b +a y \left (x \right )}{b^{3} y \left (x \right )}}}{2}\right ) \sqrt {3}\, \sqrt {\frac {4 b^{2} x y \left (x \right )+\left (3 x^{2} y \left (x \right )-2\right ) b +a y \left (x \right )}{b^{3} y \left (x \right )}}\, b -\BesselK \left (\frac {\sqrt {\frac {4 b^{3}-3 a}{b^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {4 b^{2} x y \left (x \right )+\left (3 x^{2} y \left (x \right )-2\right ) b +a y \left (x \right )}{b^{3} y \left (x \right )}}}{2}\right ) \left (b \sqrt {\frac {4 b^{3}-3 a}{b^{3}}}-2 b -3 x \right )}{\BesselI \left (\frac {\sqrt {\frac {4 b^{3}-3 a}{b^{3}}}}{2}+1, -\frac {\sqrt {3}\, \sqrt {\frac {4 b^{2} x y \left (x \right )+\left (3 x^{2} y \left (x \right )-2\right ) b +a y \left (x \right )}{b^{3} y \left (x \right )}}}{2}\right ) \sqrt {3}\, \sqrt {\frac {4 b^{2} x y \left (x \right )+\left (3 x^{2} y \left (x \right )-2\right ) b +a y \left (x \right )}{b^{3} y \left (x \right )}}\, b -\BesselI \left (\frac {\sqrt {\frac {4 b^{3}-3 a}{b^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {4 b^{2} x y \left (x \right )+\left (3 x^{2} y \left (x \right )-2\right ) b +a y \left (x \right )}{b^{3} y \left (x \right )}}}{2}\right ) \left (b \sqrt {\frac {4 b^{3}-3 a}{b^{3}}}-2 b -3 x \right )} = 0\right ]$ 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, (-1/2*I)*Sqrt[(-6*b + 3*(a + b*x*(4*b + 3*x))*y[x])/(b^3*y[x])]] - I*b*Bes
selJ[(2 + Sqrt[4 - (3*a)/b^3])/2, (-1/2*I)*Sqrt[(-6*b + 3*(a + b*x*(4*b + 3*x))*
y[x])/(b^3*y[x])]]*Sqrt[(-6*b + 3*(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, (-1/2*I)*Sqrt[(-
6*b + 3*(a + b*x*(4*b + 3*x))*y[x])/(b^3*y[x])]] + I*b*BesselY[(2 + Sqrt[4 - (3*
a)/b^3])/2, (-1/2*I)*Sqrt[(-6*b + 3*(a + b*x*(4*b + 3*x))*y[x])/(b^3*y[x])]]*Sqr
t[(-6*b + 3*(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))

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]