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y″(x)=−y′(x)(−a1−b1+1x−c1+−a2−b2+1x−c2+−a3−b3+1x−c3)−y(x)(a1b1(c1−c2)(c1−c3)x−c1+a2b2(c2−c1)(c2−c3)x−c2+a3b3(c3−c1)(c3−c2)x−c3)(x−c1)(x−c2)(x−c3) ✗ Mathematica : cpu = 96.0869 (sec), leaf count = 0 , DifferentialRoot result
\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{(\text {c1}-\unicode {f817})^2 (\text {c2}-\unicode {f817})^2 \unicode {f818}''(\unicode {f817}) (\text {c3}-\unicode {f817})^2+(\text {c1}-\unicode {f817}) (\text {c2}-\unicode {f817}) \left (\text {a1} \unicode {f817}^2+\text {a2} \unicode {f817}^2+\text {a3} \unicode {f817}^2+\text {b1} \unicode {f817}^2+\text {b2} \unicode {f817}^2+\text {b3} \unicode {f817}^2-3 \unicode {f817}^2-\text {a2} \text {c1} \unicode {f817}-\text {a3} \text {c1} \unicode {f817}-\text {b2} \text {c1} \unicode {f817}-\text {b3} \text {c1} \unicode {f817}+2 \text {c1} \unicode {f817}-\text {a1} \text {c2} \unicode {f817}-\text {a3} \text {c2} \unicode {f817}-\text {b1} \text {c2} \unicode {f817}-\text {b3} \text {c2} \unicode {f817}+2 \text {c2} \unicode {f817}-\text {a1} \text {c3} \unicode {f817}-\text {a2} \text {c3} \unicode {f817}-\text {b1} \text {c3} \unicode {f817}-\text {b2} \text {c3} \unicode {f817}+2 \text {c3} \unicode {f817}+\text {a3} \text {c1} \text {c2}+\text {b3} \text {c1} \text {c2}-\text {c1} \text {c2}+\text {a2} \text {c1} \text {c3}+\text {b2} \text {c1} \text {c3}-\text {c1} \text {c3}+\text {a1} \text {c2} \text {c3}+\text {b1} \text {c2} \text {c3}-\text {c2} \text {c3}\right ) \unicode {f818}'(\unicode {f817}) (\text {c3}-\unicode {f817})+\left (\text {a1} \text {b1} \text {c1}^2 \unicode {f817}^2+\text {a2} \text {b2} \text {c2}^2 \unicode {f817}^2+\text {a3} \text {b3} \text {c3}^2 \unicode {f817}^2-\text {a1} \text {b1} \text {c1} \text {c2} \unicode {f817}^2-\text {a2} \text {b2} \text {c1} \text {c2} \unicode {f817}^2+\text {a3} \text {b3} \text {c1} \text {c2} \unicode {f817}^2-\text {a1} \text {b1} \text {c1} \text {c3} \unicode {f817}^2+\text {a2} \text {b2} \text {c1} \text {c3} \unicode {f817}^2-\text {a3} \text {b3} \text {c1} \text {c3} \unicode {f817}^2+\text {a1} \text {b1} \text {c2} \text {c3} \unicode {f817}^2-\text {a2} \text {b2} \text {c2} \text {c3} \unicode {f817}^2-\text {a3} \text {b3} \text {c2} \text {c3} \unicode {f817}^2+\text {a1} \text {b1} \text {c1} \text {c2}^2 \unicode {f817}-\text {a2} \text {b2} \text {c1} \text {c2}^2 \unicode {f817}-\text {a3} \text {b3} \text {c1} \text {c2}^2 \unicode {f817}+\text {a1} \text {b1} \text {c1} \text {c3}^2 \unicode {f817}-\text {a2} \text {b2} \text {c1} \text {c3}^2 \unicode {f817}-\text {a3} \text {b3} \text {c1} \text {c3}^2 \unicode {f817}-\text {a1} \text {b1} \text {c2} \text {c3}^2 \unicode {f817}+\text {a2} \text {b2} \text {c2} \text {c3}^2 \unicode {f817}-\text {a3} \text {b3} \text {c2} \text {c3}^2 \unicode {f817}-\text {a1} \text {b1} \text {c1}^2 \text {c2} \unicode {f817}+\text {a2} \text {b2} \text {c1}^2 \text {c2} \unicode {f817}-\text {a3} \text {b3} \text {c1}^2 \text {c2} \unicode {f817}-\text {a1} \text {b1} \text {c1}^2 \text {c3} \unicode {f817}-\text {a2} \text {b2} \text {c1}^2 \text {c3} \unicode {f817}+\text {a3} \text {b3} \text {c1}^2 \text {c3} \unicode {f817}-\text {a1} \text {b1} \text {c2}^2 \text {c3} \unicode {f817}-\text {a2} \text {b2} \text {c2}^2 \text {c3} \unicode {f817}+\text {a3} \text {b3} \text {c2}^2 \text {c3} \unicode {f817}+2 \text {a1} \text {b1} \text {c1} \text {c2} \text {c3} \unicode {f817}+2 \text {a2} \text {b2} \text {c1} \text {c2} \text {c3} \unicode {f817}+2 \text {a3} \text {b3} \text {c1} \text {c2} \text {c3} \unicode {f817}+\text {a3} \text {b3} \text {c1}^2 \text {c2}^2+\text {a2} \text {b2} \text {c1}^2 \text {c3}^2+\text {a1} \text {b1} \text {c2}^2 \text {c3}^2-\text {a1} \text {b1} \text {c1} \text {c2} \text {c3}^2-\text {a2} \text {b2} \text {c1} \text {c2} \text {c3}^2+\text {a3} \text {b3} \text {c1} \text {c2} \text {c3}^2-\text {a1} \text {b1} \text {c1} \text {c2}^2 \text {c3}+\text {a2} \text {b2} \text {c1} \text {c2}^2 \text {c3}-\text {a3} \text {b3} \text {c1} \text {c2}^2 \text {c3}+\text {a1} \text {b1} \text {c1}^2 \text {c2} \text {c3}-\text {a2} \text {b2} \text {c1}^2 \text {c2} \text {c3}-\text {a3} \text {b3} \text {c1}^2 \text {c2} \text {c3}\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 \{(\text {c1}-\unicode {f817})^2 (\text {c2}-\unicode {f817})^2 \unicode {f818}''(\unicode {f817}) (\text {c3}-\unicode {f817})^2+(\text {c1}-\unicode {f817}) (\text {c2}-\unicode {f817}) \left (\text {a1} \unicode {f817}^2+\text {a2} \unicode {f817}^2+\text {a3} \unicode {f817}^2+\text {b1} \unicode {f817}^2+\text {b2} \unicode {f817}^2+\text {b3} \unicode {f817}^2-3 \unicode {f817}^2-\text {a2} \text {c1} \unicode {f817}-\text {a3} \text {c1} \unicode {f817}-\text {b2} \text {c1} \unicode {f817}-\text {b3} \text {c1} \unicode {f817}+2 \text {c1} \unicode {f817}-\text {a1} \text {c2} \unicode {f817}-\text {a3} \text {c2} \unicode {f817}-\text {b1} \text {c2} \unicode {f817}-\text {b3} \text {c2} \unicode {f817}+2 \text {c2} \unicode {f817}-\text {a1} \text {c3} \unicode {f817}-\text {a2} \text {c3} \unicode {f817}-\text {b1} \text {c3} \unicode {f817}-\text {b2} \text {c3} \unicode {f817}+2 \text {c3} \unicode {f817}+\text {a3} \text {c1} \text {c2}+\text {b3} \text {c1} \text {c2}-\text {c1} \text {c2}+\text {a2} \text {c1} \text {c3}+\text {b2} \text {c1} \text {c3}-\text {c1} \text {c3}+\text {a1} \text {c2} \text {c3}+\text {b1} \text {c2} \text {c3}-\text {c2} \text {c3}\right ) \unicode {f818}'(\unicode {f817}) (\text {c3}-\unicode {f817})+\left (\text {a1} \text {b1} \text {c1}^2 \unicode {f817}^2+\text {a2} \text {b2} \text {c2}^2 \unicode {f817}^2+\text {a3} \text {b3} \text {c3}^2 \unicode {f817}^2-\text {a1} \text {b1} \text {c1} \text {c2} \unicode {f817}^2-\text {a2} \text {b2} \text {c1} \text {c2} \unicode {f817}^2+\text {a3} \text {b3} \text {c1} \text {c2} \unicode {f817}^2-\text {a1} \text {b1} \text {c1} \text {c3} \unicode {f817}^2+\text {a2} \text {b2} \text {c1} \text {c3} \unicode {f817}^2-\text {a3} \text {b3} \text {c1} \text {c3} \unicode {f817}^2+\text {a1} \text {b1} \text {c2} \text {c3} \unicode {f817}^2-\text {a2} \text {b2} \text {c2} \text {c3} \unicode {f817}^2-\text {a3} \text {b3} \text {c2} \text {c3} \unicode {f817}^2+\text {a1} \text {b1} \text {c1} \text {c2}^2 \unicode {f817}-\text {a2} \text {b2} \text {c1} \text {c2}^2 \unicode {f817}-\text {a3} \text {b3} \text {c1} \text {c2}^2 \unicode {f817}+\text {a1} \text {b1} \text {c1} \text {c3}^2 \unicode {f817}-\text {a2} \text {b2} \text {c1} \text {c3}^2 \unicode {f817}-\text {a3} \text {b3} \text {c1} \text {c3}^2 \unicode {f817}-\text {a1} \text {b1} \text {c2} \text {c3}^2 \unicode {f817}+\text {a2} \text {b2} \text {c2} \text {c3}^2 \unicode {f817}-\text {a3} \text {b3} \text {c2} \text {c3}^2 \unicode {f817}-\text {a1} \text {b1} \text {c1}^2 \text {c2} \unicode {f817}+\text {a2} \text {b2} \text {c1}^2 \text {c2} \unicode {f817}-\text {a3} \text {b3} \text {c1}^2 \text {c2} \unicode {f817}-\text {a1} \text {b1} \text {c1}^2 \text {c3} \unicode {f817}-\text {a2} \text {b2} \text {c1}^2 \text {c3} \unicode {f817}+\text {a3} \text {b3} \text {c1}^2 \text {c3} \unicode {f817}-\text {a1} \text {b1} \text {c2}^2 \text {c3} \unicode {f817}-\text {a2} \text {b2} \text {c2}^2 \text {c3} \unicode {f817}+\text {a3} \text {b3} \text {c2}^2 \text {c3} \unicode {f817}+2 \text {a1} \text {b1} \text {c1} \text {c2} \text {c3} \unicode {f817}+2 \text {a2} \text {b2} \text {c1} \text {c2} \text {c3} \unicode {f817}+2 \text {a3} \text {b3} \text {c1} \text {c2} \text {c3} \unicode {f817}+\text {a3} \text {b3} \text {c1}^2 \text {c2}^2+\text {a2} \text {b2} \text {c1}^2 \text {c3}^2+\text {a1} \text {b1} \text {c2}^2 \text {c3}^2-\text {a1} \text {b1} \text {c1} \text {c2} \text {c3}^2-\text {a2} \text {b2} \text {c1} \text {c2} \text {c3}^2+\text {a3} \text {b3} \text {c1} \text {c2} \text {c3}^2-\text {a1} \text {b1} \text {c1} \text {c2}^2 \text {c3}+\text {a2} \text {b2} \text {c1} \text {c2}^2 \text {c3}-\text {a3} \text {b3} \text {c1} \text {c2}^2 \text {c3}+\text {a1} \text {b1} \text {c1}^2 \text {c2} \text {c3}-\text {a2} \text {b2} \text {c1}^2 \text {c2} \text {c3}-\text {a3} \text {b3} \text {c1}^2 \text {c2} \text {c3}\right ) \unicode {f818}(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}
✓ Maple : cpu = 1.053 (sec), leaf count = 311
{y(x)=_C1HeunG(c1−c3c1−c2,((−a3−2b1−b2+2)c1+(a3+b1−1)c2+c3(b1+b2−1))a1−(b1−1)(a2+b3)c1+(b1b3−a2b2+b3(a3−1))c2+c3(b1a2+(b2−1)a2−a3b3)c1−c2,a1+a2+b3,2−a3−b1−b2,a1−b1+1,a2−b2+1,−x+c1c1−c2)(x−c1)a1(x−c2)a2(x−c3)b3+_C2HeunG(c1−c3c1−c2,((−2a1−a3−b2+2)c1+(a1+a3−1)c2+c3(a1+b2−1))b1−(a2+b3)(a1−1)c1+(a1b3−a2b2+b3(a3−1))c2+c3(a1a2+(b2−1)a2−a3b3)c1−c2,b1+a2+b3,2−a1−a3−b2,−a1+b1+1,a2−b2+1,−x+c1c1−c2)(x−c1)b1(x−c2)a2(x−c3)b3}
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