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

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

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.177381 (sec), leaf count = 74

\[\left \{\left \{y(x)\to c_1 \text {MathieuC}\left [\frac {a^2 c^2}{2}+b^2,-\frac {1}{4} a^2 c^2,\cos ^{-1}\left (\frac {x}{a}\right )\right ]+c_2 \text {MathieuS}\left [\frac {a^2 c^2}{2}+b^2,-\frac {1}{4} a^2 c^2,\cos ^{-1}\left (\frac {x}{a}\right )\right ]\right \}\right \}\]

Maple ✓
cpu = 1.379 (sec), leaf count = 63

\[\left [y \left (x \right ) = \textit {\_C1} \MathieuC \left (\frac {a^{2} c^{2}}{2}+b^{2}, -\frac {a^{2} c^{2}}{4}, \arccos \left (\frac {x}{a}\right )\right )+\textit {\_C2} \MathieuS \left (\frac {a^{2} c^{2}}{2}+b^{2}, -\frac {a^{2} c^{2}}{4}, \arccos \left (\frac {x}{a}\right )\right )\right ]\] Mathematica raw input

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

Mathematica raw output

{{y[x] -> C[1]*MathieuC[b^2 + (a^2*c^2)/2, -1/4*(a^2*c^2), ArcCos[x/a]] + C[2]*M
athieuS[b^2 + (a^2*c^2)/2, -1/4*(a^2*c^2), ArcCos[x/a]]}}

Maple raw input

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

Maple raw output

[y(x) = _C1*MathieuC(1/2*a^2*c^2+b^2,-1/4*a^2*c^2,arccos(1/a*x))+_C2*MathieuS(1/
2*a^2*c^2+b^2,-1/4*a^2*c^2,arccos(1/a*x))]