2.1244   ODE No. 1244

\[ (n-v) (n+v+1) y(x)+2 (n+1) x y'(x)+\left (x^2-1\right ) y''(x)=0 \]

✓ Mathematica : cpu = 0.0132292 (sec), leaf count = 42

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

\[\left \{\left \{y(x)\to c_1 \left (x^2-1\right )^{-n/2} P_v^n(x)+c_2 \left (x^2-1\right )^{-n/2} Q_v^n(x)\right \}\right \}\]

✓ Maple : cpu = 0.096 (sec), leaf count = 27

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

\[y \left (x \right ) = \left (x^{2}-1\right )^{-\frac {n}{2}} \left (\operatorname {LegendreQ}\left (v , n , x\right ) c_{2} +\operatorname {LegendreP}\left (v , n , x\right ) c_{1} \right )\]