4.32.9 $(a^2-x^2)y^{″}(x)+y(x)(b^2+c^2{x}^2)-x{y}^{\prime }(x)=0$

ODE
$$(a^2-x^2)y^{″}(x)+y(x)(b^2+c^2{x}^2)-x{y}^{\prime }(x)=0$$ ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0309123 (sec), leaf count = 74

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

Maple ✓
cpu = 0.234 (sec), leaf count = 63

$$\{y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}\mathit{M}\mathit{a}\mathit{t}\mathit{h}\mathit{i}\mathit{e}\mathit{u}\mathit{C}(\frac{a^2{c}^2}{2}+b^2,-\frac{a^2{c}^2}{4},\arccos \left(\frac{x}{a}\right))+\mathit{\_}\mathit{C}\mathit{2}\mathit{M}\mathit{a}\mathit{t}\mathit{h}\mathit{i}\mathit{e}\mathit{u}\mathit{S}(\frac{a^2{c}^2}{2}+b^2,-\frac{a^2{c}^2}{4},\arccos \left(\frac{x}{a}\right))\}$$ 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))