##### 4.7.45 $$(a+b x)^2 y'(x)+y(x)^3 (a+b x)+c y(x)^2=0$$

ODE
$(a+b x)^2 y'(x)+y(x)^3 (a+b x)+c y(x)^2=0$ ODE Classiﬁcation

[_rational, _Abel]

Book solution method
Abel ODE, Second kind

Mathematica
cpu = 2.4152 (sec), leaf count = 110

$\text {Solve}\left [-\frac {c}{\sqrt {-b (a+b x)^2}}=\frac {2 \exp \left (-\frac {(b (a+b x)+c y(x))^2}{2 b y(x)^2 (a+b x)^2}\right )}{-\sqrt {2 \pi } \text {erfi}\left (\frac {b (a+b x)+c y(x)}{\sqrt {2} y(x) \sqrt {-b (a+b x)^2}}\right )+2 c_1},y(x)\right ]$

Maple
cpu = 0.144 (sec), leaf count = 153

$\left [\textit {\_C1} +\left (x +\frac {a}{b}+\frac {c \sqrt {\pi }\, \sqrt {2}\, \erf \left (\frac {\sqrt {2}\, \left (b^{2} x +a b +c y \left (x \right )\right )}{2 \sqrt {b}\, y \left (x \right ) \left (b x +a \right )}\right ) {\mathrm e}^{\frac {\left (b^{2} x +a b +c y \left (x \right )\right )^{2}}{2 y \left (x \right )^{2} \left (b x +a \right )^{2} b}}}{2 b^{\frac {3}{2}}}\right ) {\mathrm e}^{-\frac {\left (b^{2} x +b x y \left (x \right )+a b +a y \left (x \right )+c y \left (x \right )\right ) \left (b^{2} x -b x y \left (x \right )+a b -a y \left (x \right )+c y \left (x \right )\right )}{2 y \left (x \right )^{2} \left (b x +a \right )^{2} b}} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

Solve[-(c/Sqrt[-(b*(a + b*x)^2)]) == 2/(E^((b*(a + b*x) + c*y[x])^2/(2*b*(a + b*
x)^2*y[x]^2))*(2*C[1] - Sqrt[2*Pi]*Erfi[(b*(a + b*x) + c*y[x])/(Sqrt[2]*Sqrt[-(b
*(a + b*x)^2)]*y[x])])), y[x]]

Maple raw input

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

Maple raw output

[_C1+(x+a/b+1/2*c*Pi^(1/2)*2^(1/2)*erf(1/2*2^(1/2)*(b^2*x+a*b+c*y(x))/b^(1/2)/y(
x)/(b*x+a))*exp(1/2*(b^2*x+a*b+c*y(x))^2/y(x)^2/(b*x+a)^2/b)/b^(3/2))*exp(-1/2*(
b^2*x+b*x*y(x)+a*b+a*y(x)+c*y(x))*(b^2*x-b*x*y(x)+a*b-a*y(x)+c*y(x))/y(x)^2/(b*x
+a)^2/b) = 0]