4.35.24 \(a (a+1) y(x)-2 x^3 y'(x)+\left (1-x^2\right ) x^2 y''(x)=0\)

ODE
\[ a (a+1) y(x)-2 x^3 y'(x)+\left (1-x^2\right ) x^2 y''(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.28397 (sec), leaf count = 227

\[\left \{\left \{y(x)\to (-1)^{\frac {1}{4} \left (1-\sqrt {-4 a^2-4 a+1}\right )} x^{\frac {1}{2}-\frac {1}{2} \sqrt {-4 a^2-4 a+1}} \left (c_1 \, _2F_1\left (\frac {1}{4}-\frac {1}{4} \sqrt {-4 a^2-4 a+1},\frac {3}{4}-\frac {1}{4} \sqrt {-4 a^2-4 a+1};1-\frac {1}{2} \sqrt {-4 a^2-4 a+1};x^2\right )+i^{\sqrt {-4 a^2-4 a+1}} c_2 x^{\sqrt {-4 a^2-4 a+1}} \, _2F_1\left (\frac {1}{4} \left (\sqrt {-4 a^2-4 a+1}+1\right ),\frac {1}{4} \left (\sqrt {-4 a^2-4 a+1}+3\right );\frac {1}{2} \left (\sqrt {-4 a^2-4 a+1}+2\right );x^2\right )\right )\right \}\right \}\]

Maple ✓
cpu = 0.807 (sec), leaf count = 153

\[\left [y \left (x \right ) = \textit {\_C1} \hypergeom \left (\left [\frac {3}{4}-\frac {\sqrt {-4 a^{2}-4 a +1}}{4}, \frac {1}{4}-\frac {\sqrt {-4 a^{2}-4 a +1}}{4}\right ], \left [1-\frac {\sqrt {-4 a^{2}-4 a +1}}{2}\right ], x^{2}\right ) x^{\frac {1}{2}-\frac {\sqrt {-4 a^{2}-4 a +1}}{2}}+\textit {\_C2} \hypergeom \left (\left [\frac {3}{4}+\frac {\sqrt {-4 a^{2}-4 a +1}}{4}, \frac {1}{4}+\frac {\sqrt {-4 a^{2}-4 a +1}}{4}\right ], \left [1+\frac {\sqrt {-4 a^{2}-4 a +1}}{2}\right ], x^{2}\right ) x^{\frac {1}{2}+\frac {\sqrt {-4 a^{2}-4 a +1}}{2}}\right ]\] Mathematica raw input

DSolve[a*(1 + a)*y[x] - 2*x^3*y'[x] + x^2*(1 - x^2)*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (-1)^((1 - Sqrt[1 - 4*a - 4*a^2])/4)*x^(1/2 - Sqrt[1 - 4*a - 4*a^2]/2)
*(C[1]*Hypergeometric2F1[1/4 - Sqrt[1 - 4*a - 4*a^2]/4, 3/4 - Sqrt[1 - 4*a - 4*a
^2]/4, 1 - Sqrt[1 - 4*a - 4*a^2]/2, x^2] + I^Sqrt[1 - 4*a - 4*a^2]*x^Sqrt[1 - 4*
a - 4*a^2]*C[2]*Hypergeometric2F1[(1 + Sqrt[1 - 4*a - 4*a^2])/4, (3 + Sqrt[1 - 4
*a - 4*a^2])/4, (2 + Sqrt[1 - 4*a - 4*a^2])/2, x^2])}}

Maple raw input

dsolve(x^2*(-x^2+1)*diff(diff(y(x),x),x)-2*x^3*diff(y(x),x)+a*(1+a)*y(x) = 0, y(x))

Maple raw output

[y(x) = _C1*hypergeom([3/4-1/4*(-4*a^2-4*a+1)^(1/2), 1/4-1/4*(-4*a^2-4*a+1)^(1/2
)],[1-1/2*(-4*a^2-4*a+1)^(1/2)],x^2)*x^(1/2-1/2*(-4*a^2-4*a+1)^(1/2))+_C2*hyperg
eom([3/4+1/4*(-4*a^2-4*a+1)^(1/2), 1/4+1/4*(-4*a^2-4*a+1)^(1/2)],[1+1/2*(-4*a^2-
4*a+1)^(1/2)],x^2)*x^(1/2+1/2*(-4*a^2-4*a+1)^(1/2))]