ODE
\[ -(k-p) (k+p+1) y(x)+2 (1-(3-2 k) x) y'(x)+4 (1-x) x y''(x)=0 \] ODE Classification
[_Jacobi]
Book solution method
TO DO
Mathematica ✓
cpu = 0.126548 (sec), leaf count = 130
\[\left \{\left \{y(x)\to c_1 \, _2F_1\left (\frac {1}{4} \left (-2 k-\sqrt {(2 p+1)^2-8 k}+1\right ),\frac {1}{4} \left (-2 k+\sqrt {(2 p+1)^2-8 k}+1\right );\frac {1}{2};x\right )+i c_2 \sqrt {x} \, _2F_1\left (\frac {1}{4} \left (-2 k-\sqrt {(2 p+1)^2-8 k}+3\right ),\frac {1}{4} \left (-2 k+\sqrt {(2 p+1)^2-8 k}+3\right );\frac {3}{2};x\right )\right \}\right \}\]
Maple ✓
cpu = 0.081 (sec), leaf count = 65
\[ \left \{ y \left ( x \right ) = \left ( -1+x \right ) ^{{\frac {k}{2}}} \left ( {\it LegendreQ} \left ( {\frac {1}{2}\sqrt {4\,{p}^{2}-8\,k+4\,p+1}}-{\frac {1}{2}},k,\sqrt {x} \right ) {\it \_C2}+{\it LegendreP} \left ( {\frac {1}{2}\sqrt {4\,{p}^{2}-8\,k+4\,p+1}}-{\frac {1}{2}},k,\sqrt {x} \right ) {\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[-((k - p)*(1 + k + p)*y[x]) + 2*(1 - (3 - 2*k)*x)*y'[x] + 4*(1 - x)*x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Hypergeometric2F1[(1 - 2*k - Sqrt[-8*k + (1 + 2*p)^2])/4, (1 - 2*k + Sqrt[-8*k + (1 + 2*p)^2])/4, 1/2, x] + I*Sqrt[x]*C[2]*Hypergeometric2F1[(3 - 2*k - Sqrt[-8*k + (1 + 2*p)^2])/4, (3 - 2*k + Sqrt[-8*k + (1 + 2*p)^2])/4, 3/2, x]}}
Maple raw input
dsolve(4*x*(1-x)*diff(diff(y(x),x),x)+2*(1-(3-2*k)*x)*diff(y(x),x)-(k-p)*(1+k+p)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (-1+x)^(1/2*k)*(LegendreQ(1/2*(4*p^2-8*k+4*p+1)^(1/2)-1/2,k,x^(1/2))*_C2+LegendreP(1/2*(4*p^2-8*k+4*p+1)^(1/2)-1/2,k,x^(1/2))*_C1)