ODE
\[ x (\text {a0}+\text {b0} x) y'''(x)+(\text {a1}+\text {b1} x) y''(x)+x y'(x)+y(x)=f(x) \] ODE Classification
[[_3rd_order, _exact, _linear, _nonhomogeneous]]
Book solution method
TO DO
Mathematica ✓
cpu = 4.42917 (sec), leaf count = 1610
\[\left \{\left \{y(x)\to x^{-\frac {\text {a1}}{\text {a0}}} (\text {a0}+\text {b0} x)^{\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+2} \left (\text {HeunC}\left [\frac {\text {a0} \text {a1} \text {b1}-\text {a1} (\text {a0}+\text {a1}) \text {b0}}{\text {a0}^2 \text {b0}},-\frac {\text {a0}}{\text {b0}^2},\frac {\text {a1}}{\text {a0}}-1,\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} x}{\text {a0}}\right ] \left (c_1+\int _1^x\text {HeunC}\left [\frac {2 \text {b1}}{\text {b0}}-6,-\frac {\text {a0}}{\text {b0}^2},3-\frac {\text {a1}}{\text {a0}},\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[2]}{\text {a0}}\right ]dK[2] \left (c_3+\int _1^x-\frac {\text {a0} f(K[5]) (\text {a0}+\text {b0} K[5])^{-\frac {\text {a1}}{\text {a0}}+\frac {\text {b1}}{\text {b0}}-3}}{\text {b0} \text {HeunC}\left [\frac {2 \text {b1}}{\text {b0}}-6,-\frac {\text {a0}}{\text {b0}^2},3-\frac {\text {a1}}{\text {a0}},\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[5]}{\text {a0}}\right ] \text {HeunCPrime}\left [\frac {\text {a0} \text {a1} \text {b1}-\text {a1} (\text {a0}+\text {a1}) \text {b0}}{\text {a0}^2 \text {b0}},-\frac {\text {a0}}{\text {b0}^2},\frac {\text {a1}}{\text {a0}}-1,\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[5]}{\text {a0}}\right ] K[5]+\text {HeunC}\left [\frac {\text {a0} \text {a1} \text {b1}-\text {a1} (\text {a0}+\text {a1}) \text {b0}}{\text {a0}^2 \text {b0}},-\frac {\text {a0}}{\text {b0}^2},\frac {\text {a1}}{\text {a0}}-1,\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[5]}{\text {a0}}\right ] \left ((2 \text {a0}-\text {a1}) \text {HeunC}\left [\frac {2 \text {b1}}{\text {b0}}-6,-\frac {\text {a0}}{\text {b0}^2},3-\frac {\text {a1}}{\text {a0}},\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[5]}{\text {a0}}\right ]-\text {b0} \text {HeunCPrime}\left [\frac {2 \text {b1}}{\text {b0}}-6,-\frac {\text {a0}}{\text {b0}^2},3-\frac {\text {a1}}{\text {a0}},\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[5]}{\text {a0}}\right ] K[5]\right )}dK[5]\right )+\int _1^x\frac {\text {a0} f(K[3]) (\text {a0}+\text {b0} K[3])^{-\frac {\text {a1}}{\text {a0}}+\frac {\text {b1}}{\text {b0}}-3} \int _1^{K[3]}\text {HeunC}\left [\frac {2 \text {b1}}{\text {b0}}-6,-\frac {\text {a0}}{\text {b0}^2},3-\frac {\text {a1}}{\text {a0}},\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[2]}{\text {a0}}\right ]dK[2]}{\text {b0} \text {HeunC}\left [\frac {2 \text {b1}}{\text {b0}}-6,-\frac {\text {a0}}{\text {b0}^2},3-\frac {\text {a1}}{\text {a0}},\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[3]}{\text {a0}}\right ] \text {HeunCPrime}\left [\frac {\text {a0} \text {a1} \text {b1}-\text {a1} (\text {a0}+\text {a1}) \text {b0}}{\text {a0}^2 \text {b0}},-\frac {\text {a0}}{\text {b0}^2},\frac {\text {a1}}{\text {a0}}-1,\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[3]}{\text {a0}}\right ] K[3]+\text {HeunC}\left [\frac {\text {a0} \text {a1} \text {b1}-\text {a1} (\text {a0}+\text {a1}) \text {b0}}{\text {a0}^2 \text {b0}},-\frac {\text {a0}}{\text {b0}^2},\frac {\text {a1}}{\text {a0}}-1,\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[3]}{\text {a0}}\right ] \left ((2 \text {a0}-\text {a1}) \text {HeunC}\left [\frac {2 \text {b1}}{\text {b0}}-6,-\frac {\text {a0}}{\text {b0}^2},3-\frac {\text {a1}}{\text {a0}},\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[3]}{\text {a0}}\right ]-\text {b0} \text {HeunCPrime}\left [\frac {2 \text {b1}}{\text {b0}}-6,-\frac {\text {a0}}{\text {b0}^2},3-\frac {\text {a1}}{\text {a0}},\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[3]}{\text {a0}}\right ] K[3]\right )}dK[3]\right ) x^{\frac {\text {a1}}{\text {a0}}}+\text {HeunC}\left [\frac {2 \text {b1}}{\text {b0}}-6,-\frac {\text {a0}}{\text {b0}^2},3-\frac {\text {a1}}{\text {a0}},\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} x}{\text {a0}}\right ] \left (c_2-\int _1^x\text {HeunC}\left [\frac {\text {a0} \text {a1} \text {b1}-\text {a1} (\text {a0}+\text {a1}) \text {b0}}{\text {a0}^2 \text {b0}},-\frac {\text {a0}}{\text {b0}^2},\frac {\text {a1}}{\text {a0}}-1,\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[1]}{\text {a0}}\right ] K[1]^{\frac {\text {a1}}{\text {a0}}-2}dK[1] \left (c_3+\int _1^x-\frac {\text {a0} f(K[5]) (\text {a0}+\text {b0} K[5])^{-\frac {\text {a1}}{\text {a0}}+\frac {\text {b1}}{\text {b0}}-3}}{\text {b0} \text {HeunC}\left [\frac {2 \text {b1}}{\text {b0}}-6,-\frac {\text {a0}}{\text {b0}^2},3-\frac {\text {a1}}{\text {a0}},\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[5]}{\text {a0}}\right ] \text {HeunCPrime}\left [\frac {\text {a0} \text {a1} \text {b1}-\text {a1} (\text {a0}+\text {a1}) \text {b0}}{\text {a0}^2 \text {b0}},-\frac {\text {a0}}{\text {b0}^2},\frac {\text {a1}}{\text {a0}}-1,\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[5]}{\text {a0}}\right ] K[5]+\text {HeunC}\left [\frac {\text {a0} \text {a1} \text {b1}-\text {a1} (\text {a0}+\text {a1}) \text {b0}}{\text {a0}^2 \text {b0}},-\frac {\text {a0}}{\text {b0}^2},\frac {\text {a1}}{\text {a0}}-1,\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[5]}{\text {a0}}\right ] \left ((2 \text {a0}-\text {a1}) \text {HeunC}\left [\frac {2 \text {b1}}{\text {b0}}-6,-\frac {\text {a0}}{\text {b0}^2},3-\frac {\text {a1}}{\text {a0}},\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[5]}{\text {a0}}\right ]-\text {b0} \text {HeunCPrime}\left [\frac {2 \text {b1}}{\text {b0}}-6,-\frac {\text {a0}}{\text {b0}^2},3-\frac {\text {a1}}{\text {a0}},\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[5]}{\text {a0}}\right ] K[5]\right )}dK[5]\right )+\int _1^x-\frac {\text {a0} f(K[4]) (\text {a0}+\text {b0} K[4])^{-\frac {\text {a1}}{\text {a0}}+\frac {\text {b1}}{\text {b0}}-3} \int _1^{K[4]}\text {HeunC}\left [\frac {\text {a0} \text {a1} \text {b1}-\text {a1} (\text {a0}+\text {a1}) \text {b0}}{\text {a0}^2 \text {b0}},-\frac {\text {a0}}{\text {b0}^2},\frac {\text {a1}}{\text {a0}}-1,\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[1]}{\text {a0}}\right ] K[1]^{\frac {\text {a1}}{\text {a0}}-2}dK[1]}{\text {b0} \text {HeunC}\left [\frac {2 \text {b1}}{\text {b0}}-6,-\frac {\text {a0}}{\text {b0}^2},3-\frac {\text {a1}}{\text {a0}},\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[4]}{\text {a0}}\right ] \text {HeunCPrime}\left [\frac {\text {a0} \text {a1} \text {b1}-\text {a1} (\text {a0}+\text {a1}) \text {b0}}{\text {a0}^2 \text {b0}},-\frac {\text {a0}}{\text {b0}^2},\frac {\text {a1}}{\text {a0}}-1,\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[4]}{\text {a0}}\right ] K[4]+\text {HeunC}\left [\frac {\text {a0} \text {a1} \text {b1}-\text {a1} (\text {a0}+\text {a1}) \text {b0}}{\text {a0}^2 \text {b0}},-\frac {\text {a0}}{\text {b0}^2},\frac {\text {a1}}{\text {a0}}-1,\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[4]}{\text {a0}}\right ] \left ((2 \text {a0}-\text {a1}) \text {HeunC}\left [\frac {2 \text {b1}}{\text {b0}}-6,-\frac {\text {a0}}{\text {b0}^2},3-\frac {\text {a1}}{\text {a0}},\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[4]}{\text {a0}}\right ]-\text {b0} \text {HeunCPrime}\left [\frac {2 \text {b1}}{\text {b0}}-6,-\frac {\text {a0}}{\text {b0}^2},3-\frac {\text {a1}}{\text {a0}},\frac {\text {a1}}{\text {a0}}-\frac {\text {b1}}{\text {b0}}+3,0,-\frac {\text {b0} K[4]}{\text {a0}}\right ] K[4]\right )}dK[4]\right ) x^2\right )\right \}\right \}\]
Maple ✓
cpu = 2.398 (sec), leaf count = 1436
\[\text {Expression too large to display}\] Mathematica raw input
DSolve[y[x] + x*y'[x] + (a1 + b1*x)*y''[x] + x*(a0 + b0*x)*y'''[x] == f[x],y[x],x]
Mathematica raw output
{{y[x] -> ((a0 + b0*x)^(2 + a1/a0 - b1/b0)*(x^(a1/a0)*HeunC[(-(a1*(a0 + a1)*b0) + a0*a1*b1)/(a0^2*b0), -(a0/b0^2), -1 + a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*x)/a0)]*(C[1] + Inactive[Integrate][HeunC[-6 + (2*b1)/b0, -(a0/b0^2), 3 - a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[2])/a0)], {K[2], 1, x}]*(C[3] + Inactive[Integrate][-((a0*f[K[5]]*(a0 + b0*K[5])^(-3 - a1/a0 + b1/b0))/(b0*HeunC[-6 + (2*b1)/b0, -(a0/b0^2), 3 - a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[5])/a0)]*HeunCPrime[(-(a1*(a0 + a1)*b0) + a0*a1*b1)/(a0^2*b0), -(a0/b0^2), -1 + a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[5])/a0)]*K[5] + HeunC[(-(a1*(a0 + a1)*b0) + a0*a1*b1)/(a0^2*b0), -(a0/b0^2), -1 + a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[5])/a0)]*((2*a0 - a1)*HeunC[-6 + (2*b1)/b0, -(a0/b0^2), 3 - a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[5])/a0)] - b0*HeunCPrime[-6 + (2*b1)/b0, -(a0/b0^2), 3 - a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[5])/a0)]*K[5]))), {K[5], 1, x}]) + Inactive[Integrate][(a0*f[K[3]]*(a0 + b0*K[3])^(-3 - a1/a0 + b1/b0)*Inactive[Integrate][HeunC[-6 + (2*b1)/b0, -(a0/b0^2), 3 - a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[2])/a0)], {K[2], 1, K[3]}])/(b0*HeunC[-6 + (2*b1)/b0, -(a0/b0^2), 3 - a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[3])/a0)]*HeunCPrime[(-(a1*(a0 + a1)*b0) + a0*a1*b1)/(a0^2*b0), -(a0/b0^2), -1 + a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[3])/a0)]*K[3] + HeunC[(-(a1*(a0 + a1)*b0) + a0*a1*b1)/(a0^2*b0), -(a0/b0^2), -1 + a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[3])/a0)]*((2*a0 - a1)*HeunC[-6 + (2*b1)/b0, -(a0/b0^2), 3 - a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[3])/a0)] - b0*HeunCPrime[-6 + (2*b1)/b0, -(a0/b0^2), 3 - a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[3])/a0)]*K[3])), {K[3], 1, x}]) + x^2*HeunC[-6 + (2*b1)/b0, -(a0/b0^2), 3 - a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*x)/a0)]*(C[2] - Inactive[Integrate][HeunC[(-(a1*(a0 + a1)*b0) + a0*a1*b1)/(a0^2*b0), -(a0/b0^2), -1 + a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[1])/a0)]*K[1]^(-2 + a1/a0), {K[1], 1, x}]*(C[3] + Inactive[Integrate][-((a0*f[K[5]]*(a0 + b0*K[5])^(-3 - a1/a0 + b1/b0))/(b0*HeunC[-6 + (2*b1)/b0, -(a0/b0^2), 3 - a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[5])/a0)]*HeunCPrime[(-(a1*(a0 + a1)*b0) + a0*a1*b1)/(a0^2*b0), -(a0/b0^2), -1 + a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[5])/a0)]*K[5] + HeunC[(-(a1*(a0 + a1)*b0) + a0*a1*b1)/(a0^2*b0), -(a0/b0^2), -1 + a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[5])/a0)]*((2*a0 - a1)*HeunC[-6 + (2*b1)/b0, -(a0/b0^2), 3 - a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[5])/a0)] - b0*HeunCPrime[-6 + (2*b1)/b0, -(a0/b0^2), 3 - a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[5])/a0)]*K[5]))), {K[5], 1, x}]) + Inactive[Integrate][-((a0*f[K[4]]*(a0 + b0*K[4])^(-3 - a1/a0 + b1/b0)*Inactive[Integrate][HeunC[(-(a1*(a0 + a1)*b0) + a0*a1*b1)/(a0^2*b0), -(a0/b0^2), -1 + a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[1])/a0)]*K[1]^(-2 + a1/a0), {K[1], 1, K[4]}])/(b0*HeunC[-6 + (2*b1)/b0, -(a0/b0^2), 3 - a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[4])/a0)]*HeunCPrime[(-(a1*(a0 + a1)*b0) + a0*a1*b1)/(a0^2*b0), -(a0/b0^2), -1 + a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[4])/a0)]*K[4] + HeunC[(-(a1*(a0 + a1)*b0) + a0*a1*b1)/(a0^2*b0), -(a0/b0^2), -1 + a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[4])/a0)]*((2*a0 - a1)*HeunC[-6 + (2*b1)/b0, -(a0/b0^2), 3 - a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[4])/a0)] - b0*HeunCPrime[-6 + (2*b1)/b0, -(a0/b0^2), 3 - a1/a0, 3 + a1/a0 - b1/b0, 0, -((b0*K[4])/a0)]*K[4]))), {K[4], 1, x}])))/x^(a1/a0)}}
Maple raw input
dsolve(x*(b0*x+a0)*diff(diff(diff(y(x),x),x),x)+(b1*x+a1)*diff(diff(y(x),x),x)+x*diff(y(x),x)+y(x) = f(x), y(x))
Maple raw output
[y(x) = HeunC(0,(-2*a0+a1)/a0,((-b1+2*b0)*a0+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*(b0*x+a0)^(((-b1+2*b0)*a0+a1*b0)/a0/b0)*_C3+HeunC(0,(-a1+2*a0)/a0,((-b1+2*b0)*a0+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*x^((-a1+2*a0)/a0)*(b0*x+a0)^(((-b1+2*b0)*a0+a1*b0)/a0/b0)*_C2+(b0*x+a0)^(((-b1+2*b0)*a0+a1*b0)/a0/b0)*a0*(HeunC(0,(-a1+2*a0)/a0,((-b1+2*b0)*a0+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*Int(HeunC(0,(-2*a0+a1)/a0,((-b1+2*b0)*a0+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*(b0*x+a0)^(((-3*b0+b1)*a0-a1*b0)/a0/b0)*(Int(f(x),x)+_C1)*x^((-2*a0+a1)/a0)/(((-a1+2*a0)*HeunC(0,(-2*a0+a1)/a0,((-b1+2*b0)*a0+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)+x*HeunCPrime(0,(-2*a0+a1)/a0,((-b1+2*b0)*a0+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*b0)*HeunC(0,(-a1+2*a0)/a0,((-b1+2*b0)*a0+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)-HeunC(0,(-2*a0+a1)/a0,((-b1+2*b0)*a0+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*HeunCPrime(0,(-a1+2*a0)/a0,((-b1+2*b0)*a0+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*x*b0),x)*x^((-a1+2*a0)/a0)-HeunC(0,(-2*a0+a1)/a0,((-b1+2*b0)*a0+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*Int(HeunC(0,(-a1+2*a0)/a0,((-b1+2*b0)*a0+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*(b0*x+a0)^(((-3*b0+b1)*a0-a1*b0)/a0/b0)*(Int(f(x),x)+_C1)/(((-a1+2*a0)*HeunC(0,(-2*a0+a1)/a0,((-b1+2*b0)*a0+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)+x*HeunCPrime(0,(-2*a0+a1)/a0,((-b1+2*b0)*a0+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*b0)*HeunC(0,(-a1+2*a0)/a0,((-b1+2*b0)*a0+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)-HeunC(0,(-2*a0+a1)/a0,((-b1+2*b0)*a0+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*HeunCPrime(0,(-a1+2*a0)/a0,((-b1+2*b0)*a0+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*x*b0),x))]