4.2.8 $y^{\prime }(x)=a{x}^{n-1}+b{x}^{2n}+cy(x{)}^2$

ODE
$$y^{\prime }(x)=a{x}^{n-1}+b{x}^{2n}+cy(x{)}^2$$ ODE Classification

[_Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica ✓
cpu = 0.224197 (sec), leaf count = 648

$$\left\{\{y(x)\to -\frac{x^n(\sqrt{b}{c}_1(n+1)\sqrt{-(n+1{)}^2}U(-\frac{(n+1)(\sqrt{b}\sqrt{-(n+1{)}^2}n+a\sqrt{c}(n+1))}{2\sqrt{b}{(-(n+1{)}^2)}^{3/2}},\frac{n}{n+1},\frac{2\sqrt{b}\sqrt{c}{x}^{n+1}}{\sqrt{-(n+1{)}^2}})+c_1(a\sqrt{c}(n+1)+\sqrt{b}\sqrt{-(n+1{)}^2}n)U(-\frac{(n+1)(a\sqrt{c}(n+1)+\sqrt{b}\sqrt{-(n+1{)}^2}(3n+2))}{2\sqrt{b}{(-(n+1{)}^2)}^{3/2}},\frac{n}{n+1}+1,\frac{2\sqrt{b}\sqrt{c}{x}^{n+1}}{\sqrt{-(n+1{)}^2}})+\sqrt{b}(n+1)\sqrt{-(n+1{)}^2}(L_{-\frac{(n+1)(-a\sqrt{c}(n+1)-\sqrt{b}\sqrt{-(n+1{)}^2}n)}{2\sqrt{b}{(-(n+1{)}^2)}^{3/2}}}^{-\frac{1}{n+1}}\left(\frac{2\sqrt{b}\sqrt{c}{x}^{n+1}}{\sqrt{-(n+1{)}^2}}\right)+2L_{-\frac{(n+1)(-a\sqrt{c}(n+1)-\sqrt{b}\sqrt{-(n+1{)}^2}(3n+2))}{2\sqrt{b}{(-(n+1{)}^2)}^{3/2}}}^{\frac{n}{n+1}}\left(\frac{2\sqrt{b}\sqrt{c}{x}^{n+1}}{\sqrt{-(n+1{)}^2}}\right)))}{\sqrt{c}(n+1{)}^2(c_{1}U(-\frac{(n+1)(\sqrt{b}\sqrt{-(n+1{)}^2}n+a\sqrt{c}(n+1))}{2\sqrt{b}{(-(n+1{)}^2)}^{3/2}},\frac{n}{n+1},\frac{2\sqrt{b}\sqrt{c}{x}^{n+1}}{\sqrt{-(n+1{)}^2}})+L_{-\frac{(n+1)(-a\sqrt{c}(n+1)-\sqrt{b}\sqrt{-(n+1{)}^2}n)}{2\sqrt{b}{(-(n+1{)}^2)}^{3/2}}}^{-\frac{1}{n+1}}\left(\frac{2\sqrt{b}\sqrt{c}{x}^{n+1}}{\sqrt{-(n+1{)}^2}}\right))}\}\right\}$$

Maple ✓
cpu = 0.473 (sec), leaf count = 363

$$\{y\left(x\right)=-\frac{1}{2cx}(((2+n)b^{\frac{3}{2}}-i\sqrt{c}ab){\mathit{M}}_{-\frac{1}{2n+2}((-2n-2)\sqrt{b}+i\sqrt{c}a)\frac{1}{\sqrt{b}},{(2n+2)}^{-1}}\left(\frac{2i{x}^{n+1}}{n+1}\sqrt{c}\sqrt{b}\right)-2b^{3/2}\mathit{\_}\mathit{C}\mathit{1}(n+1){\mathit{W}}_{-\frac{(-2n-2)\sqrt{b}+i\sqrt{c}a}{\sqrt{b}(2n+2)},{(2n+2)}^{-1}}\left(\frac{2i\sqrt{c}\sqrt{b}{x}^{n+1}}{n+1}\right)+({\mathit{W}}_{\frac{-ia}{2n+2}\sqrt{c}\frac{1}{\sqrt{b}},{(2n+2)}^{-1}}\left(\frac{2i{x}^{n+1}}{n+1}\sqrt{c}\sqrt{b}\right)\mathit{\_}\mathit{C}\mathit{1}+{\mathit{M}}_{\frac{-ia}{2n+2}\sqrt{c}\frac{1}{\sqrt{b}},{(2n+2)}^{-1}}\left(\frac{2i{x}^{n+1}}{n+1}\sqrt{c}\sqrt{b}\right))(-b^{\frac{3}{2}}n+i\sqrt{c}(2b{x}^{n+1}+a)b))b^{-\frac{3}{2}}{({\mathit{W}}_{\frac{-ia}{2n+2}\sqrt{c}\frac{1}{\sqrt{b}},{(2n+2)}^{-1}}\left(\frac{2i{x}^{n+1}}{n+1}\sqrt{c}\sqrt{b}\right)\mathit{\_}\mathit{C}\mathit{1}+{\mathit{M}}_{\frac{-ia}{2n+2}\sqrt{c}\frac{1}{\sqrt{b}},{(2n+2)}^{-1}}\left(\frac{2i{x}^{n+1}}{n+1}\sqrt{c}\sqrt{b}\right))}^{-1}\}$$ 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[-((1 + n
)*(a*Sqrt[c]*(1 + n) + Sqrt[b]*n*Sqrt[-(1 + n)^2]))/(2*Sqrt[b]*(-(1 + n)^2)^(3/2
)), 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[-((1 + n)*(a*Sqrt[c]*(1 +
 n) + Sqrt[b]*Sqrt[-(1 + n)^2]*(2 + 3*n)))/(2*Sqrt[b]*(-(1 + n)^2)^(3/2)), 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 + n)*(-(a*Sqrt[c]*(1 + n)) - Sqrt[b]*n*Sqrt[-(1 + n
)^2]))/(2*Sqrt[b]*(-(1 + n)^2)^(3/2)), -(1 + n)^(-1), (2*Sqrt[b]*Sqrt[c]*x^(1 + 
n))/Sqrt[-(1 + n)^2]] + 2*LaguerreL[-((1 + n)*(-(a*Sqrt[c]*(1 + n)) - Sqrt[b]*Sq
rt[-(1 + n)^2]*(2 + 3*n)))/(2*Sqrt[b]*(-(1 + n)^2)^(3/2)), n/(1 + n), (2*Sqrt[b]
*Sqrt[c]*x^(1 + n))/Sqrt[-(1 + n)^2]])))/(Sqrt[c]*(1 + n)^2*(C[1]*Hypergeometric
U[-((1 + n)*(a*Sqrt[c]*(1 + n) + Sqrt[b]*n*Sqrt[-(1 + n)^2]))/(2*Sqrt[b]*(-(1 + 
n)^2)^(3/2)), n/(1 + n), (2*Sqrt[b]*Sqrt[c]*x^(1 + n))/Sqrt[-(1 + n)^2]] + Lague
rreL[-((1 + n)*(-(a*Sqrt[c]*(1 + n)) - Sqrt[b]*n*Sqrt[-(1 + n)^2]))/(2*Sqrt[b]*(
-(1 + n)^2)^(3/2)), -(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),'implicit')

Maple raw output

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