##### 4.2.8 $$y'(x)=a x^{n-1}+b x^{2 n}+c y(x)^2$$

ODE
$y'(x)=a x^{n-1}+b x^{2 n}+c y(x)^2$ ODE Classiﬁcation

[_Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.53999 (sec), leaf count = 546

$\left \{\left \{y(x)\to -\frac {x^n \left (\sqrt {b} c_1 (n+1) \sqrt {-(n+1)^2} U\left (\frac {1}{2} \left (\frac {\sqrt {c} a}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \left (a \sqrt {c} (n+1)+\sqrt {b} \sqrt {-(n+1)^2} n\right ) U\left (\frac {1}{2} \left (\frac {\sqrt {c} a}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {3 n+2}{n+1}\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+\sqrt {b} (n+1) \sqrt {-(n+1)^2} \left (L_{-\frac {\sqrt {c} a}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+2 L_{-\frac {\sqrt {c} a}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {3 n+2}{2 n+2}}^{\frac {n}{n+1}}\left (\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )\right )}{\sqrt {c} (n+1)^2 \left (L_{-\frac {\sqrt {c} a}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 U\left (\frac {1}{2} \left (\frac {\sqrt {c} a}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )}\right \}\right \}$

Maple
cpu = 0.347 (sec), leaf count = 499

$\left [y \left (x \right ) = -\frac {\left (-2 b^{\frac {3}{2}} \textit {\_C1} n -2 b^{\frac {3}{2}} \textit {\_C1} \right ) \WhittakerW \left (-\frac {i \sqrt {c}\, a -2 \sqrt {b}\, n -2 \sqrt {b}}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 b^{\frac {3}{2}} \left (\WhittakerW \left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right ) \textit {\_C1} +\WhittakerM \left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )\right ) c x}-\frac {\left (2 i \sqrt {c}\, x^{n +1} \textit {\_C1} \,b^{2}+i \sqrt {c}\, \textit {\_C1} a b -b^{\frac {3}{2}} \textit {\_C1} n \right ) \WhittakerW \left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )+\left (-i \sqrt {c}\, a b +b^{\frac {3}{2}} n +2 b^{\frac {3}{2}}\right ) \WhittakerM \left (-\frac {i \sqrt {c}\, a -2 \sqrt {b}\, n -2 \sqrt {b}}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )+\left (2 i \sqrt {c}\, x^{n +1} b^{2}+i \sqrt {c}\, a b -b^{\frac {3}{2}} n \right ) \WhittakerM \left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 b^{\frac {3}{2}} \left (\WhittakerW \left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right ) \textit {\_C1} +\WhittakerM \left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )\right ) c x}\right ]$ Mathematica raw input

DSolve[y'[x] == a*x^(-1 + n) + b*x^(2*n) + c*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> -((x^n*(Sqrt[b]*(1 + n)*Sqrt[-(1 + n)^2]*C[1]*HypergeometricU[(n/(1 +
n) + (a*Sqrt[c])/(Sqrt[b]*Sqrt[-(1 + n)^2]))/2, n/(1 + n), (2*Sqrt[b]*Sqrt[c]*x^
(1 + n))/Sqrt[-(1 + n)^2]] + (a*Sqrt[c]*(1 + n) + Sqrt[b]*n*Sqrt[-(1 + n)^2])*C[
1]*HypergeometricU[((a*Sqrt[c])/(Sqrt[b]*Sqrt[-(1 + n)^2]) + (2 + 3*n)/(1 + n))/
2, 1 + n/(1 + n), (2*Sqrt[b]*Sqrt[c]*x^(1 + n))/Sqrt[-(1 + n)^2]] + Sqrt[b]*(1 +
 n)*Sqrt[-(1 + n)^2]*(LaguerreL[-1/2*n/(1 + n) - (a*Sqrt[c])/(2*Sqrt[b]*Sqrt[-(1
 + n)^2]), -(1 + n)^(-1), (2*Sqrt[b]*Sqrt[c]*x^(1 + n))/Sqrt[-(1 + n)^2]] + 2*La
guerreL[-1/2*(a*Sqrt[c])/(Sqrt[b]*Sqrt[-(1 + n)^2]) - (2 + 3*n)/(2 + 2*n), n/(1
+ n), (2*Sqrt[b]*Sqrt[c]*x^(1 + n))/Sqrt[-(1 + n)^2]])))/(Sqrt[c]*(1 + n)^2*(C[1
]*HypergeometricU[(n/(1 + n) + (a*Sqrt[c])/(Sqrt[b]*Sqrt[-(1 + n)^2]))/2, n/(1 +
 n), (2*Sqrt[b]*Sqrt[c]*x^(1 + n))/Sqrt[-(1 + n)^2]] + LaguerreL[-1/2*n/(1 + n)
- (a*Sqrt[c])/(2*Sqrt[b]*Sqrt[-(1 + n)^2]), -(1 + n)^(-1), (2*Sqrt[b]*Sqrt[c]*x^
(1 + n))/Sqrt[-(1 + n)^2]])))}}

Maple raw input

dsolve(diff(y(x),x) = a*x^(n-1)+b*x^(2*n)+c*y(x)^2, y(x))

Maple raw output

[y(x) = -1/2*(-2*b^(3/2)*_C1*n-2*b^(3/2)*_C1)/b^(3/2)/(WhittakerW(-1/2*I*c^(1/2)
/b^(1/2)/(n+1)*a,1/2/(n+1),2*I*c^(1/2)*b^(1/2)/(n+1)*x^(n+1))*_C1+WhittakerM(-1/
2*I*c^(1/2)/b^(1/2)/(n+1)*a,1/2/(n+1),2*I*c^(1/2)*b^(1/2)/(n+1)*x^(n+1)))/c/x*Wh
ittakerW(-1/2*(I*c^(1/2)*a-2*b^(1/2)*n-2*b^(1/2))/b^(1/2)/(n+1),1/2/(n+1),2*I*c^
(1/2)*b^(1/2)/(n+1)*x^(n+1))-1/2*((2*I*c^(1/2)*x^(n+1)*_C1*b^2+I*c^(1/2)*_C1*a*b
-b^(3/2)*_C1*n)*WhittakerW(-1/2*I*c^(1/2)/b^(1/2)/(n+1)*a,1/2/(n+1),2*I*c^(1/2)*
b^(1/2)/(n+1)*x^(n+1))+(-I*c^(1/2)*a*b+b^(3/2)*n+2*b^(3/2))*WhittakerM(-1/2*(I*c
^(1/2)*a-2*b^(1/2)*n-2*b^(1/2))/b^(1/2)/(n+1),1/2/(n+1),2*I*c^(1/2)*b^(1/2)/(n+1
)*x^(n+1))+(2*I*c^(1/2)*x^(n+1)*b^2+I*c^(1/2)*a*b-b^(3/2)*n)*WhittakerM(-1/2*I*c
^(1/2)/b^(1/2)/(n+1)*a,1/2/(n+1),2*I*c^(1/2)*b^(1/2)/(n+1)*x^(n+1)))/b^(3/2)/(Wh
ittakerW(-1/2*I*c^(1/2)/b^(1/2)/(n+1)*a,1/2/(n+1),2*I*c^(1/2)*b^(1/2)/(n+1)*x^(n
+1))*_C1+WhittakerM(-1/2*I*c^(1/2)/b^(1/2)/(n+1)*a,1/2/(n+1),2*I*c^(1/2)*b^(1/2)
/(n+1)*x^(n+1)))/c/x]