4.35.31 \(y(x) \left (\text {a0}+\text {a2} x^2+\text {a4} x^4\right )+\left (1-x^2\right )^2 y''(x)-2 x \left (1-x^2\right ) y'(x)=0\)

ODE
\[ y(x) \left (\text {a0}+\text {a2} x^2+\text {a4} x^4\right )+\left (1-x^2\right )^2 y''(x)-2 x \left (1-x^2\right ) y'(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.656003 (sec), leaf count = 287

\[\left \{\left \{y(x)\to e^{i \sqrt {\text {a4}} x} \left (\frac {x+1}{x-1}\right )^{\frac {1}{2} \sqrt {-\text {a0}-\text {a2}-\text {a4}}} \left (c_2 (x-1)^{\sqrt {-\text {a0}-\text {a2}-\text {a4}}} \text {HeunC}\left [-\sqrt {-\text {a0}-\text {a2}-\text {a4}}-2 i \sqrt {\text {a4}} \left (\sqrt {-\text {a0}-\text {a2}-\text {a4}}+1\right )+\text {a0}-\text {a4},-4 i \sqrt {\text {a4}} \left (\sqrt {-\text {a0}-\text {a2}-\text {a4}}+1\right ),\sqrt {-\text {a0}-\text {a2}-\text {a4}}+1,\sqrt {-\text {a0}-\text {a2}-\text {a4}}+1,-4 i \sqrt {\text {a4}},\frac {1-x}{2}\right ]+c_1 \text {HeunC}\left [2 i \sqrt {\text {a4}} \left (\sqrt {-\text {a0}-\text {a2}-\text {a4}}-1\right )-\text {a2}-2 \text {a4},-4 i \sqrt {\text {a4}},1-\sqrt {-\text {a0}-\text {a2}-\text {a4}},\sqrt {-\text {a0}-\text {a2}-\text {a4}}+1,-4 i \sqrt {\text {a4}},\frac {1-x}{2}\right ]\right )\right \}\right \}\]

Maple ✓
cpu = 1.373 (sec), leaf count = 102

\[\left [y \left (x \right ) = \textit {\_C1} \left (x^{2}-1\right )^{\frac {\sqrt {-\mathit {a0} -\mathit {a4} -\mathit {a2}}}{2}} \HeunC \left (0, -\frac {1}{2}, \sqrt {-\mathit {a0} -\mathit {a4} -\mathit {a2}}, \frac {\mathit {a4}}{4}, \frac {1}{4}-\frac {\mathit {a0}}{4}, x^{2}\right )+\textit {\_C2} \left (x^{2}-1\right )^{\frac {\sqrt {-\mathit {a0} -\mathit {a4} -\mathit {a2}}}{2}} x \HeunC \left (0, \frac {1}{2}, \sqrt {-\mathit {a0} -\mathit {a4} -\mathit {a2}}, \frac {\mathit {a4}}{4}, \frac {1}{4}-\frac {\mathit {a0}}{4}, x^{2}\right )\right ]\] Mathematica raw input

DSolve[(a0 + a2*x^2 + a4*x^4)*y[x] - 2*x*(1 - x^2)*y'[x] + (1 - x^2)^2*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> E^(I*Sqrt[a4]*x)*((1 + x)/(-1 + x))^(Sqrt[-a0 - a2 - a4]/2)*(C[1]*Heun
C[-a2 + (2*I)*(-1 + Sqrt[-a0 - a2 - a4])*Sqrt[a4] - 2*a4, (-4*I)*Sqrt[a4], 1 - S
qrt[-a0 - a2 - a4], 1 + Sqrt[-a0 - a2 - a4], (-4*I)*Sqrt[a4], (1 - x)/2] + (-1 +
 x)^Sqrt[-a0 - a2 - a4]*C[2]*HeunC[a0 - Sqrt[-a0 - a2 - a4] - (2*I)*(1 + Sqrt[-a
0 - a2 - a4])*Sqrt[a4] - a4, (-4*I)*(1 + Sqrt[-a0 - a2 - a4])*Sqrt[a4], 1 + Sqrt
[-a0 - a2 - a4], 1 + Sqrt[-a0 - a2 - a4], (-4*I)*Sqrt[a4], (1 - x)/2])}}

Maple raw input

dsolve((-x^2+1)^2*diff(diff(y(x),x),x)-2*x*(-x^2+1)*diff(y(x),x)+(a4*x^4+a2*x^2+a0)*y(x) = 0, y(x))

Maple raw output

[y(x) = _C1*(x^2-1)^(1/2*(-a0-a4-a2)^(1/2))*HeunC(0,-1/2,(-a0-a4-a2)^(1/2),1/4*a
4,1/4-1/4*a0,x^2)+_C2*(x^2-1)^(1/2*(-a0-a4-a2)^(1/2))*x*HeunC(0,1/2,(-a0-a4-a2)^
(1/2),1/4*a4,1/4-1/4*a0,x^2)]