ODE
\[ (1-x) x (a-x) y''(x)+y'(x) \left (x^2 (\text {a0}+\text {a1}+1)+\text {a0} \text {a2}-\text {a0}+\text {a1}+x (\text {a2}+\text {a3})-\text {a3}+1\right )+\text {a0} \text {a1} (x-k) y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.60786 (sec), leaf count = 209
\[\left \{\left \{y(x)\to x^{-\frac {\text {a0} \text {a2}+\text {a1}+1}{a}} \left (c_2 x^{\frac {a+\text {a0}+\text {a3}}{a}} \text {HeunG}\left [a,\text {a0} \text {a1} k-\frac {(a (\text {a0} (\text {a2}-1)+\text {a1}+\text {a2}+1)+\text {a0} (\text {a2}-1)+\text {a1}-\text {a3}+1) (a-\text {a0} \text {a2}+\text {a0}-\text {a1}+\text {a3}-1)}{a^2},\frac {a \text {a1}+a-\text {a0} \text {a2}+\text {a0}-\text {a1}+\text {a3}-1}{a},\frac {a \text {a0}+a-\text {a0} \text {a2}+\text {a0}-\text {a1}+\text {a3}-1}{a},\frac {2 a-\text {a0} \text {a2}+\text {a0}-\text {a1}+\text {a3}-1}{a},\frac {\text {a0} \text {a2}+2 \text {a1}+\text {a2}+2}{1-a},x\right ]+c_1 x^{\frac {\text {a0} \text {a2}+\text {a1}+1}{a}} \text {HeunG}\left [a,\text {a0} \text {a1} k,\text {a0},\text {a1},\frac {\text {a0} (\text {a2}-1)+\text {a1}-\text {a3}+1}{a},\frac {\text {a0} \text {a2}+2 \text {a1}+\text {a2}+2}{1-a},x\right ]\right )\right \}\right \}\]
Maple ✓
cpu = 2.124 (sec), leaf count = 412
\[\left [y \left (x \right ) = \textit {\_C1} \left (x -1\right )^{\frac {\mathit {a0} \mathit {a2} +a +2 \mathit {a1} +\mathit {a2} +1}{a -1}} \mathit {HG}\left (a , \frac {\left (\mathit {a2}^{2}-\mathit {a2} \right ) \mathit {a0}^{2}+\left (\mathit {a2}^{2}+\left (a +3 \mathit {a1} -\mathit {a3} +1\right ) \mathit {a2} +\left (k a -k -2\right ) \mathit {a1} -a -1\right ) \mathit {a0} +\left (\mathit {a1} -\mathit {a3} +1\right ) \left (2 \mathit {a1} +a +\mathit {a2} +1\right )}{a -1}, \frac {\left (a +\mathit {a2} -1\right ) \mathit {a0} +a +2 \mathit {a1} +\mathit {a2} +1}{a -1}, \frac {a \mathit {a1} +\mathit {a0} \mathit {a2} +a +\mathit {a1} +\mathit {a2} +1}{a -1}, \frac {\left (\mathit {a2} -1\right ) \mathit {a0} +\mathit {a1} -\mathit {a3} +1}{a}, \frac {\mathit {a0} \mathit {a2} +2 a +2 \mathit {a1} +\mathit {a2}}{a -1}, x\right )+\textit {\_C2} \,x^{\frac {-\mathit {a0} \mathit {a2} +a +\mathit {a0} -\mathit {a1} +\mathit {a3} -1}{a}} \left (x -1\right )^{\frac {\mathit {a0} \mathit {a2} +a +2 \mathit {a1} +\mathit {a2} +1}{a -1}} \mathit {HG}\left (a , \frac {2 a^{4}+\left (\mathit {a1} \mathit {a0} k +2 \mathit {a0} +2 \mathit {a1} +\mathit {a2} +\mathit {a3} \right ) a^{3}+\left (\left (-\mathit {a2} +1\right ) \mathit {a0}^{2}+\left (\left (-k -\mathit {a2} \right ) \mathit {a1} +\mathit {a3} -1\right ) \mathit {a0} -\mathit {a1}^{2}+\left (\mathit {a3} -1\right ) \mathit {a1} +\mathit {a2} +\mathit {a3} \right ) a^{2}-\left (\mathit {a2} +\mathit {a3} -1\right ) \left (\left (\mathit {a2} -1\right ) \mathit {a0} +\mathit {a1} -\mathit {a3} +1\right ) a -\left (\left (\mathit {a2} -1\right ) \mathit {a0} +\mathit {a1} -\mathit {a3} +1\right )^{2}}{a^{2} \left (a -1\right )}, \frac {\left (\mathit {a1} +2\right ) a^{2}+\left (\mathit {a0} +\mathit {a2} +\mathit {a3} -1\right ) a +\left (\mathit {a2} -1\right ) \mathit {a0} +\mathit {a1} -\mathit {a3} +1}{a \left (a -1\right )}, \frac {\left (\mathit {a0} +2\right ) a^{2}+\left (\mathit {a1} +\mathit {a2} +\mathit {a3} -1\right ) a +\left (\mathit {a2} -1\right ) \mathit {a0} +\mathit {a1} -\mathit {a3} +1}{a \left (a -1\right )}, \frac {-\mathit {a0} \mathit {a2} +2 a +\mathit {a0} -\mathit {a1} +\mathit {a3} -1}{a}, \frac {\mathit {a0} \mathit {a2} +2 a +2 \mathit {a1} +\mathit {a2}}{a -1}, x\right )\right ]\] Mathematica raw input
DSolve[a0*a1*(-k + x)*y[x] + (1 - a0 + a1 + a0*a2 - a3 + (a2 + a3)*x + (1 + a0 + a1)*x^2)*y'[x] + (1 - x)*(a - x)*x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (x^((1 + a1 + a0*a2)/a)*C[1]*HeunG[a, a0*a1*k, a0, a1, (1 + a1 + a0*(-1 + a2) - a3)/a, (2 + 2*a1 + a2 + a0*a2)/(1 - a), x] + x^((a + a0 + a3)/a)*C[2]*HeunG[a, -(((1 + a1 + a0*(-1 + a2) + a*(1 + a1 + a0*(-1 + a2) + a2) - a3)*(-1 + a + a0 - a1 - a0*a2 + a3))/a^2) + a0*a1*k, (-1 + a + a0 - a1 + a*a1 - a0*a2 + a3)/a, (-1 + a + a0 + a*a0 - a1 - a0*a2 + a3)/a, (-1 + 2*a + a0 - a1 - a0*a2 + a3)/a, (2 + 2*a1 + a2 + a0*a2)/(1 - a), x])/x^((1 + a1 + a0*a2)/a)}}
Maple raw input
dsolve(x*(1-x)*(a-x)*diff(diff(y(x),x),x)+(a0*a2-a0+a1-a3+1+(a2+a3)*x+(a0+a1+1)*x^2)*diff(y(x),x)+a0*a1*(x-k)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*(x-1)^((a0*a2+a+2*a1+a2+1)/(a-1))*HeunG(a,((a2^2-a2)*a0^2+(a2^2+(a+3*a1-a3+1)*a2+(a*k-k-2)*a1-a-1)*a0+(a1-a3+1)*(2*a1+a+a2+1))/(a-1),((a+a2-1)*a0+a+2*a1+a2+1)/(a-1),(a*a1+a0*a2+a+a1+a2+1)/(a-1),((a2-1)*a0+a1-a3+1)/a,(a0*a2+2*a+2*a1+a2)/(a-1),x)+_C2*x^((-a0*a2+a+a0-a1+a3-1)/a)*(x-1)^((a0*a2+a+2*a1+a2+1)/(a-1))*HeunG(a,(2*a^4+(a0*a1*k+2*a0+2*a1+a2+a3)*a^3+((-a2+1)*a0^2+((-k-a2)*a1+a3-1)*a0-a1^2+(a3-1)*a1+a2+a3)*a^2-(a2+a3-1)*((a2-1)*a0+a1-a3+1)*a-((a2-1)*a0+a1-a3+1)^2)/a^2/(a-1),((a1+2)*a^2+(a0+a2+a3-1)*a+(a2-1)*a0+a1-a3+1)/a/(a-1),((a0+2)*a^2+(a1+a2+a3-1)*a+(a2-1)*a0+a1-a3+1)/a/(a-1),(-a0*a2+2*a+a0-a1+a3-1)/a,(a0*a2+2*a+2*a1+a2)/(a-1),x)]