4.8.37 $(1-n)x^{n-1}+x^{2n-2}+x^n{y}^{\prime }(x)+y(x{)}^2=0$

ODE
$$(1-n)x^{n-1}+x^{2n-2}+x^n{y}^{\prime }(x)+y(x{)}^2=0$$ ODE Classification

[_Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica ✗
cpu = 22.0608 (sec), leaf count = 0 , could not solve

DSolve[(1 - n)*x^(-1 + n) + x^(-2 + 2*n) + y[x]^2 + x^n*Derivative[1][y][x] == 0, y[x], x]

Maple ✓
cpu = 0.43 (sec), leaf count = 498

$$\{y\left(x\right)=-\frac{x^n}{x}(x^{\frac{3}{2}-\frac{3n}{2}+\frac{1}{2}\sqrt{n-3}\sqrt{n+1}}\mathit{\_}\mathit{C}\mathit{1}(n-1)(n-1+\sqrt{n-3}\sqrt{n+1}){}_0\text{F}{}_1(;\frac{1}{n-1}(-\sqrt{n-3}\sqrt{n+1}+2n-2);\frac{x^{1-n}}{n-1})+x^{\frac{3}{2}-\frac{3n}{2}-\frac{1}{2}\sqrt{n-3}\sqrt{n+1}}(n-1)(n-1-\sqrt{n-3}\sqrt{n+1}){}_0\text{F}{}_1(;\frac{1}{n-1}(\sqrt{n-3}\sqrt{n+1}+2n-2);\frac{x^{1-n}}{n-1})-\frac{\mathit{\_}\mathit{C}\mathit{1}}{2}{x}^{\frac{1}{2}\sqrt{n-3}\sqrt{n+1}-\frac{n}{2}+\frac{1}{2}}(-{(n-3)}^{\frac{3}{2}}{(n+1)}^{\frac{3}{2}}+(n-1)(-4+\sqrt{n+1}(n-1)\sqrt{n-3})){}_0\text{F}{}_1(;\frac{1}{n-1}(n-1-\sqrt{n-3}\sqrt{n+1});\frac{x^{1-n}}{n-1})+\frac{1}{2}{}_0\text{F}{}_1(;\frac{1}{n-1}(n-1+\sqrt{n-3}\sqrt{n+1});\frac{x^{1-n}}{n-1})(-{(n-3)}^{\frac{3}{2}}{(n+1)}^{\frac{3}{2}}+(n-1)(4+\sqrt{n+1}(n-1)\sqrt{n-3}))x^{-\frac{1}{2}\sqrt{n-3}\sqrt{n+1}-\frac{n}{2}+\frac{1}{2}}){(n-1+\sqrt{n-3}\sqrt{n+1})}^{-1}{(n-1-\sqrt{n-3}\sqrt{n+1})}^{-1}{(\mathit{\_}\mathit{C}\mathit{1}{x}^{\frac{1}{2}\sqrt{n-3}\sqrt{n+1}-\frac{n}{2}+\frac{1}{2}}{}_0\text{F}{}_1(;\frac{1}{n-1}(n-1-\sqrt{n-3}\sqrt{n+1});\frac{x^{1-n}}{n-1})+x^{-\frac{1}{2}\sqrt{n-3}\sqrt{n+1}-\frac{n}{2}+\frac{1}{2}}{}_0\text{F}{}_1(;\frac{1}{n-1}(n-1+\sqrt{n-3}\sqrt{n+1});\frac{x^{1-n}}{n-1}))}^{-1}\}$$ Mathematica raw input

DSolve[(1 - n)*x^(-1 + n) + x^(-2 + 2*n) + y[x]^2 + x^n*y'[x] == 0,y[x],x]

Mathematica raw output

DSolve[(1 - n)*x^(-1 + n) + x^(-2 + 2*n) + y[x]^2 + x^n*Derivative[1][y][x] == 0
, y[x], x]

Maple raw input

dsolve(x^n*diff(y(x),x)+x^(2*n-2)+y(x)^2+(1-n)*x^(n-1) = 0, y(x),'implicit')

Maple raw output

y(x) = -x^n*(x^(3/2-3/2*n+1/2*(n-3)^(1/2)*(n+1)^(1/2))*_C1*(n-1)*(n-1+(n-3)^(1/2
)*(n+1)^(1/2))*hypergeom([],[(-(n-3)^(1/2)*(n+1)^(1/2)+2*n-2)/(n-1)],1/(n-1)*x^(
1-n))+x^(3/2-3/2*n-1/2*(n-3)^(1/2)*(n+1)^(1/2))*(n-1)*(n-1-(n-3)^(1/2)*(n+1)^(1/
2))*hypergeom([],[((n-3)^(1/2)*(n+1)^(1/2)+2*n-2)/(n-1)],1/(n-1)*x^(1-n))-1/2*x^
(1/2*(n-3)^(1/2)*(n+1)^(1/2)-1/2*n+1/2)*_C1*(-(n-3)^(3/2)*(n+1)^(3/2)+(n-1)*(-4+
(n+1)^(1/2)*(n-1)*(n-3)^(1/2)))*hypergeom([],[(n-1-(n-3)^(1/2)*(n+1)^(1/2))/(n-1
)],1/(n-1)*x^(1-n))+1/2*hypergeom([],[1/(n-1)*(n-1+(n-3)^(1/2)*(n+1)^(1/2))],1/(
n-1)*x^(1-n))*(-(n-3)^(3/2)*(n+1)^(3/2)+(n-1)*(4+(n+1)^(1/2)*(n-1)*(n-3)^(1/2)))
*x^(-1/2*(n-3)^(1/2)*(n+1)^(1/2)-1/2*n+1/2))/x/(n-1+(n-3)^(1/2)*(n+1)^(1/2))/(n-
1-(n-3)^(1/2)*(n+1)^(1/2))/(_C1*x^(1/2*(n-3)^(1/2)*(n+1)^(1/2)-1/2*n+1/2)*hyperg
eom([],[(n-1-(n-3)^(1/2)*(n+1)^(1/2))/(n-1)],1/(n-1)*x^(1-n))+x^(-1/2*(n-3)^(1/2
)*(n+1)^(1/2)-1/2*n+1/2)*hypergeom([],[1/(n-1)*(n-1+(n-3)^(1/2)*(n+1)^(1/2))],1/
(n-1)*x^(1-n)))