ODE No. 1572

6.39 ODE No. 1572

${(x^{2} - 1)}^{2} d 4 y (x) + 10 x (x^{2} - 1) \frac{d^{3}}{d x^{3}} y (x) + (24 x^{2} - 8 - 2 (μ (μ + 1) + ν (ν + 1)) (x^{2} - 1)) \frac{d^{2}}{d x^{2}} y (x) - 6 x (μ (μ + 1) + ν (ν + 1) - 2) \frac{d}{d x} y (x) + ({(μ (μ + 1) - ν (ν + 1))}^{2} - 2 μ (μ + 1) - 2 ν (ν + 1)) y (x) = 0$

Mathematica: cpu = 93.272844 (sec), leaf count = 174 $\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{(\mu -\nu -1) (\mu -\nu +1) (\mu +\nu ) (\mu +\nu +2) \unicode {f818}(\unicode {f817})-6 \unicode {f817} \left (\mu ^2+\mu +\nu ^2+\nu -2\right ) \unicode {f818}'(\unicode {f817})-2 \left (-12 \unicode {f817}^3+\mu ^2 \unicode {f817}^2+\nu ^2 \unicode {f817}^2+\mu \unicode {f817}^2+\nu \unicode {f817}^2-\mu ^2-\nu ^2-\mu -\nu +4\right ) \unicode {f818}''(\unicode {f817})+\left (10 \unicode {f817}^3-10 \unicode {f817}\right ) \unicode {f818}^{(3)}(\unicode {f817})+\left (\unicode {f817}^4-2 \unicode {f817}^2+1\right ) \unicode {f818}^{(4)}(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2,\unicode {f818}''(0)=c_3,\unicode {f818}^{(3)}(0)=c_4\right \},\langle \langle \rangle \rangle \right )(x)\right \}\right \}$

Maple: cpu = 0.203 (sec), leaf count = 37 ${y (x) =_C 1 L e g e n d r e P (ν, x) L e g e n d r e P (μ, x) +_C 2 L e g e n d r e P (ν, x) L e g e n d r e Q (μ, x) +_C 3 L e g e n d r e Q (ν, x) L e g e n d r e P (μ, x) +_C 4 L e g e n d r e Q (ν, x) L e g e n d r e Q (μ, x)}$

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