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 = 0.224197 (sec), leaf count = 648

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

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

\[ \left \{ y \relax (x ) =-{\frac {1}{2\,cx} \left (\left (\left (2+n \right ) {b}^{{\frac {3}{2}}}-i\sqrt {c}ab \right ) {{\sl M}_{-{\frac {1}{2\,n+2} \left (\left (-2\,n-2 \right ) \sqrt {b}+i\sqrt {c}a \right ) {\frac {1}{\sqrt {b}}}},\, \left (2\,n+2 \right ) ^{-1}}\left ({\frac {2\,i{x}^{n+1}}{n+1}\sqrt {c}\sqrt {b}}\right )}-2\,{b}^{3/2}{\it \_C1}\, \left (n+1 \right ) {{\sl W}_{-{\frac { \left (-2\,n-2 \right ) \sqrt {b}+i\sqrt {c}a}{\sqrt {b} \left (2\,n+2 \right ) }},\, \left (2\,n+2 \right ) ^{-1}}\left ({\frac {2\,i\sqrt {c}\sqrt {b}{x}^{n+1}}{n+1}}\right )}+ \left ({{\sl W}_{{\frac {-ia}{2\,n+2}\sqrt {c}{\frac {1}{\sqrt {b}}}},\, \left (2\,n+2 \right ) ^{-1}}\left ({\frac {2\,i{x}^{n+1}}{n+1}\sqrt {c}\sqrt {b}}\right )}{\it \_C1}+{{\sl M}_{{\frac {-ia}{2\,n+2}\sqrt {c}{\frac {1}{\sqrt {b}}}},\, \left (2\,n+2 \right ) ^{-1}}\left ({\frac {2\,i{x}^{n+1}}{n+1}\sqrt {c}\sqrt {b}}\right )} \right ) \left (-{b}^{{\frac {3}{2}}}n+i\sqrt {c} \left (2\,b{x}^{n+1}+a \right ) b \right ) \right ) {b}^{-{\frac {3}{2}}} \left ({{\sl W}_{{\frac {-ia}{2\,n+2}\sqrt {c}{\frac {1}{\sqrt {b}}}},\, \left (2\,n+2 \right ) ^{-1}}\left ({\frac {2\,i{x}^{n+1}}{n+1}\sqrt {c}\sqrt {b}}\right )}{\it \_C1}+{{\sl M}_{{\frac {-ia}{2\,n+2}\sqrt {c}{\frac {1}{\sqrt {b}}}},\, \left (2\,n+2 \right ) ^{-1}}\left ({\frac {2\,i{x}^{n+1}}{n+1}\sqrt {c}\sqrt {b}}\right )} \right ) ^{-1}} \right \} \] 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[-((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