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  ODE No. 1403

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 = 14.3933 (sec), leaf count = 316 

DSolve[Derivative[2][y][x] == -((((a1*b1*(c1 - c2)*(c1 - c3))/(-c1 + x) + (a2*b2*(-c1 + c2)*(c2 - c3))/(-c2 + x) + (a3*b3*(-c1 + c3)*(-c2 + c3))/(-c3 + x))*y[x])/((-c1 + x)*(-c2 + x)*(-c3 + x))) - ((1 - a1 - b1)/(-c1 + x) + (1 - a2 - b2)/(-c2 + x) + (1 - a3 - b3)/(-c3 + x))*Derivative[1][y][x],y[x],x]

{{y(x)→c1(x−c1)a1(x−c2)a2(x−c3)b3HeunG[c1−c3c1−c2,a1(c1(−a3−2b1−b2+2)+c2(a3+b1−1)+c3(b1+b2−1))+c2(−a2b2+(a3−1)b3+b1b3)+c3(a2b1+a2(b2−1)−a3b3)+(1−b1)c1(a2+b3)c1−c2,a1+a2+b3,−a3−b1−b2+2,a1−b1+1,a2−b2+1,c1−xc1−c2]+c2(x−c2)a2(x−c1)b1(x−c3)b3HeunG[c1−c3c1−c2,c2(a1b3−a2b2+(a3−1)b3)+c3(a1a2+a2(b2−1)−a3b3)+(1−a1)c1(a2+b3)+b1(c1(−2a1−a3−b2+2)+c2(a1+a3−1)+c3(a1+b2−1))c1−c2,a2+b1+b3,−a1−a3−b2+2,−a1+b1+1,a2−b2+1,c1−xc1−c2]}} ✓ Maple : cpu = 0.615 (sec), leaf count = 298 

dsolve(diff(diff(y(x),x),x) = -((1-a1-b1)/(x-c1)+(1-a2-b2)/(x-c2)+(1-a3-b3)/(x-c3))*diff(y(x),x)-1/(x-c1)/(x-c2)/(x-c3)*(a1*b1*(c1-c3)*(c1-c2)/(x-c1)+a2*b2*(c2-c1)*(c2-c3)/(x-c2)+a3*b3*(c3-c2)*(c3-c1)/(x-c3))*y(x),y(x))

y(x)=(x−c2)a2(x−c3)b3((x−c1)a1HG(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+(b1a2+(b2−1)a2−a3b3)c3c1−c2,a1+b3+a2,2−a3−b1−b2,a1−b1+1,a2−b2+1,−x+c1c1−c2)c1+(x−c1)b1HG(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+(a1a2+(b2−1)a2−a3b3)c3c1−c2,b1+b3+a2,2−a1−a3−b2,−a1+b1+1,a2−b2+1,−x+c1c1−c2)c2)

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