ODE
\[ x^n y'(x)=a^2 x^{2 n-2}+b^2 y(x)^2 \] ODE Classification
[[_homogeneous, `class G`], _Riccati]
Book solution method
Riccati ODE, Generalized ODE
Mathematica ✓
cpu = 0.0685813 (sec), leaf count = 143
\[\left \{\left \{y(x)\to \frac {x^{n-1} \left (c_1 \left (a b \sqrt {\frac {(n-1)^2-4 a^2 b^2}{a^2 b^2}}+n-1\right )+\left (-a b \sqrt {\frac {(n-1)^2-4 a^2 b^2}{a^2 b^2}}+n-1\right ) x^{a b \sqrt {\frac {(n-1)^2-4 a^2 b^2}{a^2 b^2}}}\right )}{2 b^2 \left (x^{a b \sqrt {\frac {(n-1)^2}{a^2 b^2}-4}}+c_1\right )}\right \}\right \}\]
Maple ✓
cpu = 0.02 (sec), leaf count = 72
\[ \left \{ \ln \left ( x \right ) -{\it \_C1}-2\,{\frac {1}{\sqrt {4\,{a}^{2}{b}^{2}-{n}^{2}+2\,n-1}}\arctan \left ( {\frac {2\,{b}^{2}y \left ( x \right ) {x}^{1-n}-n+1}{\sqrt {4\,{a}^{2}{b}^{2}-{n}^{2}+2\,n-1}}} \right ) }=0 \right \} \] Mathematica raw input
DSolve[x^n*y'[x] == a^2*x^(-2 + 2*n) + b^2*y[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> (x^(-1 + n)*((-1 - a*b*Sqrt[(-4*a^2*b^2 + (-1 + n)^2)/(a^2*b^2)] + n)*x^(a*b*Sqrt[(-4*a^2*b^2 + (-1 + n)^2)/(a^2*b^2)]) + (-1 + a*b*Sqrt[(-4*a^2*b^2 + (-1 + n)^2)/(a^2*b^2)] + n)*C[1]))/(2*b^2*(x^(a*b*Sqrt[-4 + (-1 + n)^2/(a^2*b^2)]) + C[1]))}}
Maple raw input
dsolve(x^n*diff(y(x),x) = a^2*x^(2*n-2)+b^2*y(x)^2, y(x),'implicit')
Maple raw output
ln(x)-_C1-2/(4*a^2*b^2-n^2+2*n-1)^(1/2)*arctan((2*b^2*y(x)*x^(1-n)-n+1)/(4*a^2*b^2-n^2+2*n-1)^(1/2)) = 0