4.32.8 \((a+b x) y'(x)+c y(x)+\left (1-x^2\right ) y''(x)=0\)

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

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.156769 (sec), leaf count = 184

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

Maple ✓
cpu = 0.087 (sec), leaf count = 134

\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\mbox {$_2$F$_1$}(-{\frac {1}{2}}-{\frac {b}{2}}-{\frac {1}{2}\sqrt {{b}^{2}+2\,b+4\,c+1}},-{\frac {1}{2}}-{\frac {b}{2}}+{\frac {1}{2}\sqrt {{b}^{2}+2\,b+4\,c+1}};\,{\frac {a}{2}}-{\frac {b}{2}};\,{\frac {1}{2}}+{\frac {x}{2}})}+{\it \_C2}\, \left ( {\frac {1}{2}}+{\frac {x}{2}} \right ) ^{1-{\frac {a}{2}}+{\frac {b}{2}}}{\mbox {$_2$F$_1$}({\frac {1}{2}}-{\frac {1}{2}\sqrt {{b}^{2}+2\,b+4\,c+1}}-{\frac {a}{2}},{\frac {1}{2}}+{\frac {1}{2}\sqrt {{b}^{2}+2\,b+4\,c+1}}-{\frac {a}{2}};\,2-{\frac {a}{2}}+{\frac {b}{2}};\,{\frac {1}{2}}+{\frac {x}{2}})} \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

dsolve((-x^2+1)*diff(diff(y(x),x),x)+(b*x+a)*diff(y(x),x)+c*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = _C1*hypergeom([-1/2-1/2*b-1/2*(b^2+2*b+4*c+1)^(1/2), -1/2-1/2*b+1/2*(b^2+
2*b+4*c+1)^(1/2)],[1/2*a-1/2*b],1/2+1/2*x)+_C2*(1/2+1/2*x)^(1-1/2*a+1/2*b)*hyper
geom([1/2-1/2*(b^2+2*b+4*c+1)^(1/2)-1/2*a, 1/2+1/2*(b^2+2*b+4*c+1)^(1/2)-1/2*a],
[2-1/2*a+1/2*b],1/2+1/2*x)