\[ x^{-a-1} y(x)^2-x^a+y'(x)=0 \] ✓ Mathematica : cpu = 0.0655547 (sec), leaf count = 230
\[\left \{\left \{y(x)\to \frac {x^{a+1} \left (c_1 \left (\frac {1}{2} x^{-\frac {a}{2}-\frac {1}{2}} \Gamma (a+1) \left (I_{a-1}\left (2 \sqrt {x}\right )+I_{a+1}\left (2 \sqrt {x}\right )\right )-\frac {1}{2} a x^{-\frac {a}{2}-1} \Gamma (a+1) I_a\left (2 \sqrt {x}\right )\right )-\frac {1}{2} (-1)^{-a} a x^{-\frac {a}{2}-1} \Gamma (1-a) I_{-a}\left (2 \sqrt {x}\right )+\frac {1}{2} (-1)^{-a} x^{-\frac {a}{2}-\frac {1}{2}} \Gamma (1-a) \left (I_{-a-1}\left (2 \sqrt {x}\right )+I_{1-a}\left (2 \sqrt {x}\right )\right )\right )}{c_1 x^{-a/2} \Gamma (a+1) I_a\left (2 \sqrt {x}\right )+(-1)^{-a} x^{-a/2} \Gamma (1-a) I_{-a}\left (2 \sqrt {x}\right )}\right \}\right \}\]
✓ Maple : cpu = 0.085 (sec), leaf count = 54
\[ \left \{ y \left ( x \right ) ={{x}^{a+1} \left ( -{{\sl K}_{a+1}\left (2\,\sqrt {x}\right )}{\it \_C1}+{{\sl I}_{a+1}\left (2\,\sqrt {x}\right )} \right ) {\frac {1}{\sqrt {x}}} \left ( {{\sl K}_{a}\left (2\,\sqrt {x}\right )}{\it \_C1}+{{\sl I}_{a}\left (2\,\sqrt {x}\right )} \right ) ^{-1}} \right \} \]
Hand solution
\begin {align} y^{\prime }+x^{-a-1}y^{2}-x^{a} & =0\nonumber \\ y^{\prime } & =x^{a}-x^{-a-1}y^{2}\nonumber \\ & =P\left ( x\right ) +Q\left ( x\right ) y+R\left ( x\right ) y^{2} \tag {1} \end {align}
This is Ricatti first order non-linear ODE. Using standard transformation\[ y=-\frac {u^{\prime }}{uR\left ( x\right ) }=x^{a+1}\frac {u^{\prime }}{u}\]
Hence
\[ y^{\prime }=\left ( a+1\right ) x^{a}\frac {u^{\prime }}{u}+x^{a+1}\frac {u^{\prime \prime }}{u}-x^{a+1}\frac {\left ( u^{\prime }\right ) ^{2}}{u^{2}}\]
Comparing to (1) gives
\begin {align} x^{a}-x^{-a-1}y^{2} & =\left ( a+1\right ) x^{a}\frac {u^{\prime }}{u}+x^{a+1}\frac {u^{\prime \prime }}{u}-x^{a+1}\frac {\left ( u^{\prime }\right ) ^{2}}{u^{2}}\nonumber \\ x^{a}-x^{-a-1}\left ( x^{a+1}\frac {u^{\prime }}{u}\right ) ^{2} & =\left ( a+1\right ) x^{a}\frac {u^{\prime }}{u}+x^{a+1}\frac {u^{\prime \prime }}{u}-x^{a+1}\frac {\left ( u^{\prime }\right ) ^{2}}{u^{2}}\nonumber \\ 1-\frac {x^{-a-1}}{x^{a}}x^{2a+2}\frac {\left ( u^{\prime }\right ) ^{2}}{u^{2}} & =\left ( a+1\right ) \frac {u^{\prime }}{u}+x\frac {u^{\prime \prime }}{u}-x\frac {\left ( u^{\prime }\right ) ^{2}}{u^{2}}\nonumber \\ 1-x\frac {\left ( u^{\prime }\right ) ^{2}}{u^{2}} & =\left ( a+1\right ) \frac {u^{\prime }}{u}+x\frac {u^{\prime \prime }}{u}-x\frac {\left ( u^{\prime }\right ) ^{2}}{u^{2}}\nonumber \\ 1 & =\left ( a+1\right ) \frac {u^{\prime }}{u}+x\frac {u^{\prime \prime }}{u}\nonumber \\ xu^{\prime \prime }+\left ( 1+a\right ) u^{\prime }-u & =0 \tag {2} \end {align}
This is Bessel like second order linear ODE. The solution is
\[ u=C_{1}\frac {1}{\sqrt {x^{a}}}\operatorname {BesselI}\left ( a,2\sqrt {x}\right ) +C_{2}\frac {1}{\sqrt {x^{a}}}\operatorname {BesselK}\left ( a,2\sqrt {x}\right ) \]
But \begin {align*} \frac {d}{dx}\frac {1}{\sqrt {x^{a}}}\operatorname {BesselI}\left ( a,2\sqrt {x}\right ) & =\frac {1}{\sqrt {x^{1+a}}}\operatorname {BesselI}\left ( 1+a,2\sqrt {x}\right ) \\ \frac {d}{dx}\frac {1}{\sqrt {x^{a}}}\operatorname {BesselI}\left ( a,2\sqrt {x}\right ) & =-\frac {1}{\sqrt {x^{1+a}}}\operatorname {BesselK}\left ( 1+a,2\sqrt {x}\right ) \end {align*}
Hence
\[ u^{\prime }=C_{1}\frac {1}{\sqrt {x^{1+a}}}\operatorname {BesselI}\left ( 1+a,2\sqrt {x}\right ) -C_{2}\frac {1}{\sqrt {x^{1+a}}}\operatorname {BesselK}\left ( 1+a,2\sqrt {x}\right ) \]
And from \(y=x^{a+1}\frac {u^{\prime }}{u}\)
\[ y=x^{1+a}\frac {C_{1}\frac {1}{\sqrt {x^{1+a}}}\operatorname {BesselI}\left ( 1+a,2\sqrt {x}\right ) -C_{2}\frac {1}{\sqrt {x^{1+a}}}\operatorname {BesselK}\left ( 1+a,2\sqrt {x}\right ) }{C_{1}\frac {1}{\sqrt {x^{a}}}\operatorname {BesselI}\left ( a,2\sqrt {x}\right ) +C_{2}\frac {1}{\sqrt {x^{a}}}\operatorname {BesselK}\left ( a,2\sqrt {x}\right ) }\]
Let \(C=\frac {C_{2}}{C_{1}}\) hence
\[ y=x^{1+a}\frac {\frac {1}{\sqrt {x^{1+a}}}\operatorname {BesselI}\left ( 1+a,2\sqrt {x}\right ) -C\frac {1}{\sqrt {x^{1+a}}}\operatorname {BesselK}\left ( 1+a,2\sqrt {x}\right ) }{\frac {1}{\sqrt {x^{a}}}\operatorname {BesselI}\left ( a,2\sqrt {x}\right ) +C\frac {1}{\sqrt {x^{a}}}\operatorname {BesselK}\left ( a,2\sqrt {x}\right ) }\]
Or
\begin {align*} y & =x^{1+a}\frac {x^{-\frac {1}{2}}\operatorname {BesselI}\left ( 1+a,2\sqrt {x}\right ) -Cx^{-\frac {1}{2}}\operatorname {BesselK}\left ( 1+a,2\sqrt {x}\right ) }{\operatorname {BesselI}\left ( a,2\sqrt {x}\right ) +C\operatorname {BesselK}\left ( a,2\sqrt {x}\right ) }\\ & =\frac {x^{\frac {1}{2}+a}\operatorname {BesselI}\left ( 1+a,2\sqrt {x}\right ) -Cx^{\frac {1}{2}+a}\operatorname {BesselK}\left ( 1+a,2\sqrt {x}\right ) }{\operatorname {BesselI}\left ( a,2\sqrt {x}\right ) +C\operatorname {BesselK}\left ( a,2\sqrt {x}\right ) } \end {align*}
Verification
eq:=diff(y(x),x)+x^(-a-1)*y(x)^2-x^a = 0; num:=x^(1/2+a)*BesselI(1+a,2*sqrt(x))-_C1*x^(1/2+a)*BesselK(1+a,2*sqrt(x)); den:=BesselI(a,2*sqrt(x))+_C1*BesselK(a,2*sqrt(x)); my_sol:=num/den; odetest(y(x)=my_sol,eq); 0