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y″(x)=−y(x)(a2(x2+1)2+m2−n(n+1)(x2+1))(x2+1)2−2xy′(x)x2+1 ✗ Mathematica : cpu = 2.29995 (sec), leaf count = 0 , DifferentialRoot result
\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\unicode {f818}''(\unicode {f817}) \left (\unicode {f817}^2+1\right )^2+2 \unicode {f817} \unicode {f818}'(\unicode {f817}) \left (\unicode {f817}^2+1\right )+\left (a^2 \unicode {f817}^4+2 a^2 \unicode {f817}^2-n^2 \unicode {f817}^2-n \unicode {f817}^2+a^2+m^2-n^2-n\right ) \unicode {f818}(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\unicode {f818}''(\unicode {f817}) \left (\unicode {f817}^2+1\right )^2+2 \unicode {f817} \unicode {f818}'(\unicode {f817}) \left (\unicode {f817}^2+1\right )+\left (a^2 \unicode {f817}^4+2 a^2 \unicode {f817}^2-n^2 \unicode {f817}^2-n \unicode {f817}^2+a^2+m^2-n^2-n\right ) \unicode {f818}(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}
✓ Maple : cpu = 0.492 (sec), leaf count = 96
{y(x)=_C1(x2+1)m2HeunC(0,−12,m,−a24,14+a24+m24−n24−n4,−x2)+_C2(x2+1)m2xHeunC(0,12,m,−a24,14+a24+m24−n24−n4,−x2)}
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