4.8.40 $x^{2n}{y}^{\prime }(x)=-n{x}^{n-1}+x^{n}y(x)(x^{2n}y(x{)}^2-3x^{n}y(x)+1)+1$

ODE
$$x^{2n}{y}^{\prime }(x)=-n{x}^{n-1}+x^{n}y(x)(x^{2n}y(x{)}^2-3x^{n}y(x)+1)+1$$ ODE Classification

[_Abel]

Book solution method
Abel ODE, First kind

Mathematica ✓
cpu = 0.250927 (sec), leaf count = 254

$$\{\{y(x)\to x^{-n}-\frac{e^{\frac{2x^{1-n}}{n-1}}}{\sqrt{c_1-\frac{2x(\frac{4^{\frac{n+1}{n-1}}x{\left(\frac{x^{1-n}}{1-n}\right)}^{\frac{2}{n-1}}\mathrm{\Gamma }(-\frac{2}{n-1},-\frac{4x^{1-n}}{n-1})}{n-1}+e^{\frac{4x^{1-n}}{n-1}}{x}^n)}{n+1}}}\},\{y(x)\to \frac{e^{\frac{2x^{1-n}}{n-1}}}{\sqrt{c_1-\frac{2x(\frac{4^{\frac{n+1}{n-1}}x{\left(\frac{x^{1-n}}{1-n}\right)}^{\frac{2}{n-1}}\mathrm{\Gamma }(-\frac{2}{n-1},-\frac{4x^{1-n}}{n-1})}{n-1}+e^{\frac{4x^{1-n}}{n-1}}{x}^n)}{n+1}}}+x^{-n}\}\}$$

Maple ✓
cpu = 0.095 (sec), leaf count = 962

$$\{y\left(x\right)-1{\mathrm{e}}^{2\frac{x}{(n-1)x^n}}\frac{1}{\sqrt{\mathit{\_}\mathit{C}\mathit{1}-2\frac{1}{1-n}{2}^{-2\frac{n}{1-n}-2{(1-n)}^{-1}}{\left({(1-n)}^{-1}\right)}^{-\frac{n}{1-n}-{(1-n)}^{-1}}(-\frac{(n-1)(1-n)}{(n+1)(n-3)}{2}^{-3+2\frac{n}{1-n}+2{(1-n)}^{-1}+2{(n-1)}^{-1}}{x}^{-\frac{n^2}{1-n}+{(1-n)}^{-1}-1+n}{\left({(1-n)}^{-1}\right)}^{\frac{n}{1-n}+{(1-n)}^{-1}}(-4\frac{x^{1-n}{n}^2}{1-n}+8\frac{x^{1-n}n}{1-n}-4\frac{x^{1-n}}{1-n}+2n-2){\left(\frac{x^{1-n}}{1-n}\right)}^{{(n-1)}^{-1}}{\mathrm{e}}^{2\frac{x^{1-n}}{n-1}}{\mathit{M}}_{-\frac{n}{n-1},-{(n-1)}^{-1}+1/2}(-4\frac{x^{1-n}}{n-1})+\frac{(n-1)(1-n)}{(n+1)(n-3)}{2}^{-1+2\frac{n}{1-n}+2{(1-n)}^{-1}+2{(n-1)}^{-1}}{x}^{-\frac{n^2}{1-n}+{(1-n)}^{-1}-1+n}{\left({(1-n)}^{-1}\right)}^{\frac{n}{1-n}+{(1-n)}^{-1}}{\left(\frac{x^{1-n}}{1-n}\right)}^{{(n-1)}^{-1}}{\mathrm{e}}^{2\frac{x^{1-n}}{n-1}}{\mathit{M}}_{-{(n-1)}^{-1},-{(n-1)}^{-1}+1/2}(-4\frac{x^{1-n}}{n-1}))}}-{\left(x^n\right)}^{-1}=0,y\left(x\right)+1{\mathrm{e}}^{2\frac{x}{(n-1)x^n}}\frac{1}{\sqrt{\mathit{\_}\mathit{C}\mathit{1}-2\frac{1}{1-n}{2}^{-2\frac{n}{1-n}-2{(1-n)}^{-1}}{\left({(1-n)}^{-1}\right)}^{-\frac{n}{1-n}-{(1-n)}^{-1}}(-\frac{(n-1)(1-n)}{(n+1)(n-3)}{2}^{-3+2\frac{n}{1-n}+2{(1-n)}^{-1}+2{(n-1)}^{-1}}{x}^{-\frac{n^2}{1-n}+{(1-n)}^{-1}-1+n}{\left({(1-n)}^{-1}\right)}^{\frac{n}{1-n}+{(1-n)}^{-1}}(-4\frac{x^{1-n}{n}^2}{1-n}+8\frac{x^{1-n}n}{1-n}-4\frac{x^{1-n}}{1-n}+2n-2){\left(\frac{x^{1-n}}{1-n}\right)}^{{(n-1)}^{-1}}{\mathrm{e}}^{2\frac{x^{1-n}}{n-1}}{\mathit{M}}_{-\frac{n}{n-1},-{(n-1)}^{-1}+1/2}(-4\frac{x^{1-n}}{n-1})+\frac{(n-1)(1-n)}{(n+1)(n-3)}{2}^{-1+2\frac{n}{1-n}+2{(1-n)}^{-1}+2{(n-1)}^{-1}}{x}^{-\frac{n^2}{1-n}+{(1-n)}^{-1}-1+n}{\left({(1-n)}^{-1}\right)}^{\frac{n}{1-n}+{(1-n)}^{-1}}{\left(\frac{x^{1-n}}{1-n}\right)}^{{(n-1)}^{-1}}{\mathrm{e}}^{2\frac{x^{1-n}}{n-1}}{\mathit{M}}_{-{(n-1)}^{-1},-{(n-1)}^{-1}+1/2}(-4\frac{x^{1-n}}{n-1}))}}-{\left(x^n\right)}^{-1}=0\}$$ Mathematica raw input

DSolve[x^(2*n)*y'[x] == 1 - n*x^(-1 + n) + x^n*y[x]*(1 - 3*x^n*y[x] + x^(2*n)*y[x]^2),y[x],x]

Mathematica raw output

{{y[x] -> x^(-n) - E^((2*x^(1 - n))/(-1 + n))/Sqrt[C[1] - (2*x*(E^((4*x^(1 - n))
/(-1 + n))*x^n + (4^((1 + n)/(-1 + n))*x*(x^(1 - n)/(1 - n))^(2/(-1 + n))*Gamma[
-2/(-1 + n), (-4*x^(1 - n))/(-1 + n)])/(-1 + n)))/(1 + n)]}, {y[x] -> x^(-n) + E
^((2*x^(1 - n))/(-1 + n))/Sqrt[C[1] - (2*x*(E^((4*x^(1 - n))/(-1 + n))*x^n + (4^
((1 + n)/(-1 + n))*x*(x^(1 - n)/(1 - n))^(2/(-1 + n))*Gamma[-2/(-1 + n), (-4*x^(
1 - n))/(-1 + n)])/(-1 + n)))/(1 + n)]}}

Maple raw input

dsolve(x^(2*n)*diff(y(x),x) = 1-n*x^(n-1)+x^n*y(x)*(1-3*x^n*y(x)+x^(2*n)*y(x)^2), y(x),'implicit')

Maple raw output

y(x)+exp(2*x/(n-1)/(x^n))/(_C1-2*2^(-2*n/(1-n)-2/(1-n))*(1/(1-n))^(-n/(1-n)-1/(1
-n))/(1-n)*(-2^(-3+2*n/(1-n)+2/(1-n)+2/(n-1))*(n-1)/(n+1)*x^(-n^2/(1-n)+1/(1-n)-
1+n)*(1/(1-n))^(n/(1-n)+1/(1-n))*(-4*x^(1-n)/(1-n)*n^2+8*n*x^(1-n)/(1-n)-4*x^(1-
n)/(1-n)+2*n-2)/(n-3)*(1-n)*(x^(1-n)/(1-n))^(1/(n-1))*exp(2*x^(1-n)/(n-1))*Whitt
akerM(-1/(n-1)*n,-1/(n-1)+1/2,-4*x^(1-n)/(n-1))+2^(-1+2*n/(1-n)+2/(1-n)+2/(n-1))
*(n-1)/(n+1)*x^(-n^2/(1-n)+1/(1-n)-1+n)*(1/(1-n))^(n/(1-n)+1/(1-n))/(n-3)*(1-n)*
(x^(1-n)/(1-n))^(1/(n-1))*exp(2*x^(1-n)/(n-1))*WhittakerM(-1/(n-1),-1/(n-1)+1/2,
-4*x^(1-n)/(n-1))))^(1/2)-1/(x^n) = 0, y(x)-exp(2*x/(n-1)/(x^n))/(_C1-2*2^(-2*n/
(1-n)-2/(1-n))*(1/(1-n))^(-n/(1-n)-1/(1-n))/(1-n)*(-2^(-3+2*n/(1-n)+2/(1-n)+2/(n
-1))*(n-1)/(n+1)*x^(-n^2/(1-n)+1/(1-n)-1+n)*(1/(1-n))^(n/(1-n)+1/(1-n))*(-4*x^(1
-n)/(1-n)*n^2+8*n*x^(1-n)/(1-n)-4*x^(1-n)/(1-n)+2*n-2)/(n-3)*(1-n)*(x^(1-n)/(1-n
))^(1/(n-1))*exp(2*x^(1-n)/(n-1))*WhittakerM(-1/(n-1)*n,-1/(n-1)+1/2,-4*x^(1-n)/
(n-1))+2^(-1+2*n/(1-n)+2/(1-n)+2/(n-1))*(n-1)/(n+1)*x^(-n^2/(1-n)+1/(1-n)-1+n)*(
1/(1-n))^(n/(1-n)+1/(1-n))/(n-3)*(1-n)*(x^(1-n)/(1-n))^(1/(n-1))*exp(2*x^(1-n)/(
n-1))*WhittakerM(-1/(n-1),-1/(n-1)+1/2,-4*x^(1-n)/(n-1))))^(1/2)-1/(x^n) = 0