4.8.35 $x^n{y}^{\prime }(x)=x^{2n-1}-y(x{)}^2$

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

[_Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica ✓
cpu = 0.0675225 (sec), leaf count = 222

$$\left\{\{y(x)\to \frac{x^{n-1}(c_1(-1{)}^{n+1}\sqrt{x}\mathrm{\Gamma }(n)I_{n-2}\left(2\sqrt{x}\right)+c_1(-1{)}^{n+1}\mathrm{\Gamma }(n)I_{n-1}\left(2\sqrt{x}\right)+c_1(-1{)}^{n}n\mathrm{\Gamma }(n)I_{n-1}\left(2\sqrt{x}\right)+c_1(-1{)}^{n+1}\sqrt{x}\mathrm{\Gamma }(n)I_n\left(2\sqrt{x}\right)-(n-1)\mathrm{\Gamma }(2-n)I_{1-n}\left(2\sqrt{x}\right)+\sqrt{x}\mathrm{\Gamma }(2-n)I_{2-n}\left(2\sqrt{x}\right)+\sqrt{x}\mathrm{\Gamma }(2-n)I_{-n}\left(2\sqrt{x}\right))}{2(c_1(-1{)}^{n+1}\mathrm{\Gamma }(n)I_{n-1}\left(2\sqrt{x}\right)+\mathrm{\Gamma }(2-n)I_{1-n}\left(2\sqrt{x}\right))}\}\right\}$$

Maple ✓
cpu = 0.084 (sec), leaf count = 52

$$\{y\left(x\right)=x^n(-{\mathit{K}}_n\left(2\sqrt{x}\right)\mathit{\_}\mathit{C}\mathit{1}+{\mathit{I}}_n\left(2\sqrt{x}\right))\frac{1}{\sqrt{x}}{({\mathit{K}}_{n-1}\left(2\sqrt{x}\right)\mathit{\_}\mathit{C}\mathit{1}+{\mathit{I}}_{n-1}\left(2\sqrt{x}\right))}^{-1}\}$$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (x^(-1 + n)*(-((-1 + n)*BesselI[1 - n, 2*Sqrt[x]]*Gamma[2 - n]) + Sqrt
[x]*BesselI[2 - n, 2*Sqrt[x]]*Gamma[2 - n] + Sqrt[x]*BesselI[-n, 2*Sqrt[x]]*Gamm
a[2 - n] + (-1)^(1 + n)*Sqrt[x]*BesselI[-2 + n, 2*Sqrt[x]]*C[1]*Gamma[n] + (-1)^
(1 + n)*BesselI[-1 + n, 2*Sqrt[x]]*C[1]*Gamma[n] + (-1)^n*n*BesselI[-1 + n, 2*Sq
rt[x]]*C[1]*Gamma[n] + (-1)^(1 + n)*Sqrt[x]*BesselI[n, 2*Sqrt[x]]*C[1]*Gamma[n])
)/(2*(BesselI[1 - n, 2*Sqrt[x]]*Gamma[2 - n] + (-1)^(1 + n)*BesselI[-1 + n, 2*Sq
rt[x]]*C[1]*Gamma[n]))}}

Maple raw input

dsolve(x^n*diff(y(x),x) = x^(2*n-1)-y(x)^2, y(x),'implicit')

Maple raw output

y(x) = 1/x^(1/2)*(-BesselK(n,2*x^(1/2))*_C1+BesselI(n,2*x^(1/2)))*x^n/(BesselK(n
-1,2*x^(1/2))*_C1+BesselI(n-1,2*x^(1/2)))