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

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

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.244223 (sec), leaf count = 158

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

Maple ✓
cpu = 0.14 (sec), leaf count = 143

\[ \left \{ y \left ( x \right ) = \left ( {x}^{2}-1 \right ) ^{{\frac {a}{2}}+{\frac {b}{2}}} \left ( {x}^{2}-1 \right ) \left ( {\it \_C2}\,{x}^{1-a}{\mbox {$_2$F$_1$}({\frac {5}{4}}+{\frac {b}{4}}+{\frac {1}{4}\sqrt {{b}^{2}+2\,b+4\,c+1}},{\frac {5}{4}}+{\frac {b}{4}}-{\frac {1}{4}\sqrt {{b}^{2}+2\,b+4\,c+1}};\,-{\frac {a}{2}}+{\frac {3}{2}};\,{x}^{2})}+{\it \_C1}\,{\mbox {$_2$F$_1$}({\frac {3}{4}}+{\frac {a}{2}}+{\frac {b}{4}}-{\frac {1}{4}\sqrt {{b}^{2}+2\,b+4\,c+1}},{\frac {3}{4}}+{\frac {a}{2}}+{\frac {b}{4}}+{\frac {1}{4}\sqrt {{b}^{2}+2\,b+4\,c+1}};\,{\frac {1}{2}}+{\frac {a}{2}};\,{x}^{2})} \right ) \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> C[1]*Hypergeometric2F1[(-1 - b - Sqrt[1 + 2*b + b^2 + 4*c])/4, (-1 - b
 + Sqrt[1 + 2*b + b^2 + 4*c])/4, (1 + a)/2, x^2] + I^(1 - a)*x^(1 - a)*C[2]*Hype
rgeometric2F1[(1 - 2*a - b - Sqrt[1 + 2*b + b^2 + 4*c])/4, (1 - 2*a - b + Sqrt[1
 + 2*b + b^2 + 4*c])/4, (3 - a)/2, x^2]}}

Maple raw input

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

Maple raw output

y(x) = (x^2-1)^(1/2*a+1/2*b)*(x^2-1)*(_C2*x^(1-a)*hypergeom([5/4+1/4*b+1/4*(b^2+
2*b+4*c+1)^(1/2), 5/4+1/4*b-1/4*(b^2+2*b+4*c+1)^(1/2)],[-1/2*a+3/2],x^2)+_C1*hyp
ergeom([3/4+1/2*a+1/4*b-1/4*(b^2+2*b+4*c+1)^(1/2), 3/4+1/2*a+1/4*b+1/4*(b^2+2*b+
4*c+1)^(1/2)],[1/2+1/2*a],x^2))