##### 4.4.36 $$x y'(x)=a x^m-b y(x)-c x^n y(x)^2$$

ODE
$x y'(x)=a x^m-b y(x)-c x^n y(x)^2$ ODE Classiﬁcation

[_rational, _Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.547201 (sec), leaf count = 1293

$\left \{\left \{y(x)\to \frac {x^{-n} \left ((-1)^{\frac {n}{m+n}} \sqrt {a} \sqrt {c} m (m+n)^{\frac {2 n}{m+n}} x^{m+n} I_{-\frac {b+m}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \Gamma \left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}}+(-1)^{\frac {n}{m+n}} \sqrt {a} \sqrt {c} n (m+n)^{\frac {2 n}{m+n}} x^{m+n} I_{-\frac {b+m}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \Gamma \left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}}+(-1)^{\frac {n}{m+n}} \sqrt {a} \sqrt {c} m (m+n)^{\frac {2 n}{m+n}} x^{m+n} I_{\frac {n-b}{m+n}+1}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \Gamma \left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}}+(-1)^{\frac {n}{m+n}} \sqrt {a} \sqrt {c} n (m+n)^{\frac {2 n}{m+n}} x^{m+n} I_{\frac {n-b}{m+n}+1}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \Gamma \left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}}+(-1)^{\frac {b}{m+n}} \sqrt {a} \sqrt {c} (m+n)^{\frac {2 b}{m+n}+1} x^{m+n} I_{\frac {b+m}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \Gamma \left (\frac {b+m}{m+n}\right ) \left ((m+n)^2\right )^{\frac {n}{m+n}}+(-1)^{\frac {b}{m+n}} \sqrt {a} \sqrt {c} m (m+n)^{\frac {2 b}{m+n}} x^{m+n} I_{\frac {b-n}{m+n}-1}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \Gamma \left (\frac {b+m}{m+n}\right ) \left ((m+n)^2\right )^{\frac {n}{m+n}}+(-1)^{\frac {b}{m+n}} \sqrt {a} \sqrt {c} n (m+n)^{\frac {2 b}{m+n}} x^{m+n} I_{\frac {b-n}{m+n}-1}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \Gamma \left (\frac {b+m}{m+n}\right ) \left ((m+n)^2\right )^{\frac {n}{m+n}}+(-1)^{\frac {n}{m+n}+1} b (m+n)^{\frac {2 n}{m+n}} \sqrt {x^{m+n}} I_{\frac {n-b}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \Gamma \left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}+\frac {1}{2}}+(-1)^{\frac {n}{m+n}} n (m+n)^{\frac {2 n}{m+n}} \sqrt {x^{m+n}} I_{\frac {n-b}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \Gamma \left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}+\frac {1}{2}}+(-1)^{\frac {b}{m+n}} (n-b) (m+n)^{\frac {2 b}{m+n}} \sqrt {x^{m+n}} I_{\frac {b-n}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \Gamma \left (\frac {b+m}{m+n}\right ) \left ((m+n)^2\right )^{\frac {n}{m+n}+\frac {1}{2}}\right )}{2 c \sqrt {(m+n)^2} \sqrt {x^{m+n}} \left ((-1)^{\frac {n}{m+n}} (m+n)^{\frac {2 n}{m+n}} I_{\frac {n-b}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \Gamma \left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}}+(-1)^{\frac {b}{m+n}} (m+n)^{\frac {2 b}{m+n}} I_{\frac {b-n}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \Gamma \left (\frac {b+m}{m+n}\right ) \left ((m+n)^2\right )^{\frac {n}{m+n}}\right )}\right \}\right \}$

Maple
cpu = 0.243 (sec), leaf count = 174

$\left [y \left (x \right ) = -\frac {\left (\BesselY \left (\frac {b +m}{n +m}, \frac {2 \sqrt {-c a}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right ) \textit {\_C1} +\BesselJ \left (\frac {b +m}{n +m}, \frac {2 \sqrt {-c a}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right )\right ) x^{\frac {n}{2}+\frac {m}{2}} \sqrt {-c a}\, x^{1-n}}{\left (\BesselY \left (\frac {b -n}{n +m}, \frac {2 \sqrt {-c a}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right ) \textit {\_C1} +\BesselJ \left (\frac {b -n}{n +m}, \frac {2 \sqrt {-c a}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right )\right ) c x}\right ]$ Mathematica raw input

DSolve[x*y'[x] == a*x^m - b*y[x] - c*x^n*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> ((-1)^(b/(m + n))*Sqrt[a]*Sqrt[c]*(m + n)^(1 + (2*b)/(m + n))*((m + n)
^2)^(n/(m + n))*x^(m + n)*BesselI[(b + m)/(m + n), (2*Sqrt[a]*Sqrt[c]*Sqrt[x^(m
+ n)])/Sqrt[(m + n)^2]]*C[1]*Gamma[(b + m)/(m + n)] + (-1)^(b/(m + n))*(-b + n)*
(m + n)^((2*b)/(m + n))*((m + n)^2)^(1/2 + n/(m + n))*Sqrt[x^(m + n)]*BesselI[(b
 - n)/(m + n), (2*Sqrt[a]*Sqrt[c]*Sqrt[x^(m + n)])/Sqrt[(m + n)^2]]*C[1]*Gamma[(
b + m)/(m + n)] + (-1)^(b/(m + n))*Sqrt[a]*Sqrt[c]*m*(m + n)^((2*b)/(m + n))*((m
 + n)^2)^(n/(m + n))*x^(m + n)*BesselI[-1 + (b - n)/(m + n), (2*Sqrt[a]*Sqrt[c]*
Sqrt[x^(m + n)])/Sqrt[(m + n)^2]]*C[1]*Gamma[(b + m)/(m + n)] + (-1)^(b/(m + n))
*Sqrt[a]*Sqrt[c]*n*(m + n)^((2*b)/(m + n))*((m + n)^2)^(n/(m + n))*x^(m + n)*Bes
selI[-1 + (b - n)/(m + n), (2*Sqrt[a]*Sqrt[c]*Sqrt[x^(m + n)])/Sqrt[(m + n)^2]]*
C[1]*Gamma[(b + m)/(m + n)] + (-1)^(n/(m + n))*Sqrt[a]*Sqrt[c]*m*(m + n)^((2*n)/
(m + n))*((m + n)^2)^(b/(m + n))*x^(m + n)*BesselI[-((b + m)/(m + n)), (2*Sqrt[a
]*Sqrt[c]*Sqrt[x^(m + n)])/Sqrt[(m + n)^2]]*Gamma[(-b + m + 2*n)/(m + n)] + (-1)
^(n/(m + n))*Sqrt[a]*Sqrt[c]*n*(m + n)^((2*n)/(m + n))*((m + n)^2)^(b/(m + n))*x
^(m + n)*BesselI[-((b + m)/(m + n)), (2*Sqrt[a]*Sqrt[c]*Sqrt[x^(m + n)])/Sqrt[(m
 + n)^2]]*Gamma[(-b + m + 2*n)/(m + n)] + (-1)^(1 + n/(m + n))*b*(m + n)^((2*n)/
(m + n))*((m + n)^2)^(1/2 + b/(m + n))*Sqrt[x^(m + n)]*BesselI[(-b + n)/(m + n),
 (2*Sqrt[a]*Sqrt[c]*Sqrt[x^(m + n)])/Sqrt[(m + n)^2]]*Gamma[(-b + m + 2*n)/(m +
n)] + (-1)^(n/(m + n))*n*(m + n)^((2*n)/(m + n))*((m + n)^2)^(1/2 + b/(m + n))*S
qrt[x^(m + n)]*BesselI[(-b + n)/(m + n), (2*Sqrt[a]*Sqrt[c]*Sqrt[x^(m + n)])/Sqr
t[(m + n)^2]]*Gamma[(-b + m + 2*n)/(m + n)] + (-1)^(n/(m + n))*Sqrt[a]*Sqrt[c]*m
*(m + n)^((2*n)/(m + n))*((m + n)^2)^(b/(m + n))*x^(m + n)*BesselI[1 + (-b + n)/
(m + n), (2*Sqrt[a]*Sqrt[c]*Sqrt[x^(m + n)])/Sqrt[(m + n)^2]]*Gamma[(-b + m + 2*
n)/(m + n)] + (-1)^(n/(m + n))*Sqrt[a]*Sqrt[c]*n*(m + n)^((2*n)/(m + n))*((m + n
)^2)^(b/(m + n))*x^(m + n)*BesselI[1 + (-b + n)/(m + n), (2*Sqrt[a]*Sqrt[c]*Sqrt
[x^(m + n)])/Sqrt[(m + n)^2]]*Gamma[(-b + m + 2*n)/(m + n)])/(2*c*Sqrt[(m + n)^2
]*x^n*Sqrt[x^(m + n)]*((-1)^(b/(m + n))*(m + n)^((2*b)/(m + n))*((m + n)^2)^(n/(
m + n))*BesselI[(b - n)/(m + n), (2*Sqrt[a]*Sqrt[c]*Sqrt[x^(m + n)])/Sqrt[(m + n
)^2]]*C[1]*Gamma[(b + m)/(m + n)] + (-1)^(n/(m + n))*(m + n)^((2*n)/(m + n))*((m
 + n)^2)^(b/(m + n))*BesselI[(-b + n)/(m + n), (2*Sqrt[a]*Sqrt[c]*Sqrt[x^(m + n)
])/Sqrt[(m + n)^2]]*Gamma[(-b + m + 2*n)/(m + n)]))}}

Maple raw input

dsolve(x*diff(y(x),x) = a*x^m-b*y(x)-c*x^n*y(x)^2, y(x))

Maple raw output

[y(x) = -(BesselY((b+m)/(n+m),2*(-c*a)^(1/2)*x^(1/2*n+1/2*m)/(n+m))*_C1+BesselJ(
(b+m)/(n+m),2*(-c*a)^(1/2)*x^(1/2*n+1/2*m)/(n+m)))*x^(1/2*n+1/2*m)*(-c*a)^(1/2)/
(BesselY((b-n)/(n+m),2*(-c*a)^(1/2)*x^(1/2*n+1/2*m)/(n+m))*_C1+BesselJ((b-n)/(n+
m),2*(-c*a)^(1/2)*x^(1/2*n+1/2*m)/(n+m)))*x^(1-n)/c/x]