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 = 2.50611 (sec), leaf count = 110

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

Maple
cpu = 0.047 (sec), leaf count = 153

\[ \left \{ {\frac {1}{2} \left (\left ({\it Erf} \left ({\frac {\sqrt {2} \left (cy \relax (x ) +b \left (bx+a \right ) \right ) }{2\, \left (bx+a \right ) y \relax (x ) }{\frac {1}{\sqrt {b}}}} \right ) {{\rm e}^{{\frac { \left (cy \relax (x ) +b \left (bx+a \right ) \right ) ^{2}}{2\, \left (y \relax (x ) \right ) ^{2} \left (bx+a \right ) ^{2}b}}}}\sqrt {\pi }\sqrt {2}bc+2\, \left (bx+a \right ) {b}^{3/2} \right ) {{\rm e}^{-{\frac { \left (\left (-bx-a+c \right ) y \relax (x ) +b \left (bx+a \right ) \right ) \left (\left (bx+a+c \right ) y \relax (x ) +b \left (bx+a \right ) \right ) }{2\, \left (y \relax (x ) \right ) ^{2} \left (bx+a \right ) ^{2}b}}}}+2\,{\it \_C1}\,{b}^{5/2} \right ) {b}^{-{\frac {5}{2}}}}=0 \right \} \] 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),'implicit')

Maple raw output

1/2*((erf(1/2*2^(1/2)*(c*y(x)+b*(b*x+a))/b^(1/2)/y(x)/(b*x+a))*exp(1/2*(c*y(x)+b
*(b*x+a))^2/y(x)^2/(b*x+a)^2/b)*Pi^(1/2)*2^(1/2)*b*c+2*(b*x+a)*b^(3/2))*exp(-1/2
*((-b*x-a+c)*y(x)+b*(b*x+a))*((b*x+a+c)*y(x)+b*(b*x+a))/y(x)^2/(b*x+a)^2/b)+2*_C
1*b^(5/2))/b^(5/2) = 0