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 = 0.0309123 (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 = 0.234 (sec), leaf count = 63

\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\it MathieuC} \left ( {\frac {{a}^{2}{c}^{2}}{2}}+{b}^{2},-{\frac {{a}^{2}{c}^{2}}{4}},\arccos \left ( {\frac {x}{a}} \right ) \right ) +{\it \_C2}\,{\it MathieuS} \left ( {\frac {{a}^{2}{c}^{2}}{2}}+{b}^{2},-{\frac {{a}^{2}{c}^{2}}{4}},\arccos \left ( {\frac {x}{a}} \right ) \right ) \right \} \] 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, -(a^2*c^2)/4, ArcCos[x/a]] + C[2]*Mat
hieuS[b^2 + (a^2*c^2)/2, -(a^2*c^2)/4, 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),'implicit')

Maple raw output

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