4.8.36 $y(x)(x^{n-1}+y(x))+x^n{y}^{\prime }(x)+x^2=0$

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

[_Riccati]

Book solution method
Change of Variable, new dependent variable

Mathematica ✓
cpu = 0.201479 (sec), leaf count = 470

$$\left\{\{y(x)\to -\frac{x^{n-1}(c_1{\left(x^{2n}\right)}^{\frac{1}{n}}\mathrm{\Gamma }\left(\frac{4-3n}{4-2n}\right)(-J_{\frac{n}{2(n-2)}-1}\left(\frac{{\left(x^{2n}\right)}^{\frac{1}{n}-\frac{1}{2}}}{2-n}\right))+c_1{\left(x^{2n}\right)}^{\frac{1}{n}}\mathrm{\Gamma }\left(\frac{4-3n}{4-2n}\right)J_{\frac{n}{2(n-2)}+1}\left(\frac{{\left(x^{2n}\right)}^{\frac{1}{n}-\frac{1}{2}}}{2-n}\right)+c_{1}n\sqrt{x^{2n}}\mathrm{\Gamma }\left(\frac{4-3n}{4-2n}\right)J_{\frac{n}{2(n-2)}}\left(\frac{{\left(x^{2n}\right)}^{\frac{1}{n}-\frac{1}{2}}}{2-n}\right)+{\left(x^{2n}\right)}^{\frac{1}{n}}\mathrm{\Gamma }\left(\frac{n-4}{2(n-2)}\right)J_{\frac{n-4}{2(n-2)}}\left(\frac{{\left(x^{2n}\right)}^{\frac{1}{n}-\frac{1}{2}}}{2-n}\right)-{\left(x^{2n}\right)}^{\frac{1}{n}}\mathrm{\Gamma }\left(\frac{n-4}{2(n-2)}\right)J_{\frac{n}{4-2n}-1}\left(\frac{{\left(x^{2n}\right)}^{\frac{1}{n}-\frac{1}{2}}}{2-n}\right)+n\sqrt{x^{2n}}\mathrm{\Gamma }\left(\frac{n-4}{2(n-2)}\right)J_{\frac{n}{4-2n}}\left(\frac{{\left(x^{2n}\right)}^{\frac{1}{n}-\frac{1}{2}}}{2-n}\right))}{2\sqrt{x^{2n}}(c_1\mathrm{\Gamma }\left(\frac{4-3n}{4-2n}\right)J_{\frac{n}{2(n-2)}}\left(\frac{{\left(x^{2n}\right)}^{\frac{1}{n}-\frac{1}{2}}}{2-n}\right)+\mathrm{\Gamma }\left(\frac{n-4}{2(n-2)}\right)J_{\frac{n}{4-2n}}\left(\frac{{\left(x^{2n}\right)}^{\frac{1}{n}-\frac{1}{2}}}{2-n}\right))}\}\right\}$$

Maple ✓
cpu = 0.64 (sec), leaf count = 218

$$\{y\left(x\right)=\frac{x^n}{x(n-4)}(\frac{x^{-3n+4}\mathit{\_}\mathit{C}\mathit{1}(n-4)}{3}{}_0\text{F}{}_1(;\frac{5n-8}{2n-4};-\frac{x^{-2n+4}}{4{(n-2)}^2})+(n-\frac{4}{3})(-{}_0\text{F}{}_1(;\frac{3n-4}{2n-4};-\frac{x^{-2n+4}}{4{(n-2)}^2})n(n-4)\mathit{\_}\mathit{C}\mathit{1}{x}^{-n}+x^{-2n+4}{}_0\text{F}{}_1(;\frac{3n-8}{2n-4};-\frac{x^{-2n+4}}{4{(n-2)}^2}))){(n-\frac{4}{3})}^{-1}{(\mathit{\_}\mathit{C}\mathit{1}{x}^{-n}{}_0\text{F}{}_1(;\frac{3n-4}{2n-4};-\frac{x^{-2n+4}}{4{(n-2)}^2})+{}_0\text{F}{}_1(;\frac{n-4}{2n-4};-\frac{x^{-2n+4}}{4{(n-2)}^2}))}^{-1}\}$$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> -(x^(-1 + n)*(n*Sqrt[x^(2*n)]*BesselJ[n/(2*(-2 + n)), (x^(2*n))^(-1/2 
+ n^(-1))/(2 - n)]*C[1]*Gamma[(4 - 3*n)/(4 - 2*n)] - (x^(2*n))^n^(-1)*BesselJ[-1
 + n/(2*(-2 + n)), (x^(2*n))^(-1/2 + n^(-1))/(2 - n)]*C[1]*Gamma[(4 - 3*n)/(4 - 
2*n)] + (x^(2*n))^n^(-1)*BesselJ[1 + n/(2*(-2 + n)), (x^(2*n))^(-1/2 + n^(-1))/(
2 - n)]*C[1]*Gamma[(4 - 3*n)/(4 - 2*n)] + (x^(2*n))^n^(-1)*BesselJ[(-4 + n)/(2*(
-2 + n)), (x^(2*n))^(-1/2 + n^(-1))/(2 - n)]*Gamma[(-4 + n)/(2*(-2 + n))] + n*Sq
rt[x^(2*n)]*BesselJ[n/(4 - 2*n), (x^(2*n))^(-1/2 + n^(-1))/(2 - n)]*Gamma[(-4 + 
n)/(2*(-2 + n))] - (x^(2*n))^n^(-1)*BesselJ[-1 + n/(4 - 2*n), (x^(2*n))^(-1/2 + 
n^(-1))/(2 - n)]*Gamma[(-4 + n)/(2*(-2 + n))]))/(2*Sqrt[x^(2*n)]*(BesselJ[n/(2*(
-2 + n)), (x^(2*n))^(-1/2 + n^(-1))/(2 - n)]*C[1]*Gamma[(4 - 3*n)/(4 - 2*n)] + B
esselJ[n/(4 - 2*n), (x^(2*n))^(-1/2 + n^(-1))/(2 - n)]*Gamma[(-4 + n)/(2*(-2 + n
))]))}}

Maple raw input

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

Maple raw output

y(x) = (1/3*x^(-3*n+4)*_C1*hypergeom([],[(5*n-8)/(2*n-4)],-1/4/(n-2)^2*x^(-2*n+4
))*(n-4)+(n-4/3)*(-hypergeom([],[(3*n-4)/(2*n-4)],-1/4/(n-2)^2*x^(-2*n+4))*n*(n-
4)*_C1*x^(-n)+x^(-2*n+4)*hypergeom([],[(3*n-8)/(2*n-4)],-1/4/(n-2)^2*x^(-2*n+4))
))*x^n/(n-4/3)/(n-4)/(_C1*x^(-n)*hypergeom([],[(3*n-4)/(2*n-4)],-1/4/(n-2)^2*x^(
-2*n+4))+hypergeom([],[(n-4)/(2*n-4)],-1/4/(n-2)^2*x^(-2*n+4)))/x