ODE
\[ (\text {a1}-x) (\text {a2}-x) (\text {a3}-x) y''(x)+y'(x) \left (\text {b0}+\text {b1} x+\text {b2} x^2\right )+y(x) (\text {c0}+\text {c1} x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 61.6258 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{(-\text {c0}-\unicode {f817} \text {c1}) \unicode {f818}(\unicode {f817})+\left (-\text {b2} \unicode {f817}^2-\text {b1} \unicode {f817}-\text {b0}\right ) \unicode {f818}'(\unicode {f817})-(\text {a1}-\unicode {f817}) (\text {a2}-\unicode {f817}) (\text {a3}-\unicode {f817}) \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}\]
Maple ✓
cpu = 1.22 (sec), leaf count = 1123
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\it HeunG} \left ( {\frac {{\it a1}-{\it a3}}{{\it a1}-{\it a2}}},{\frac {-{\it a1}\,{\it c1}-{\it c0}}{{\it a1}-{\it a2}}},-{\frac {{\it b2}}{2}}-{\frac {1}{2}}+{\frac {1}{2}\sqrt {{{\it b2}}^{2}+2\,{\it b2}+4\,{\it c1}+1}},{1 \left ( \left ( \left ( {\it a1}\,{\it b2}+{\it b2}\,{\it a2}+{\it b1} \right ) {\it a3}-{\it a1}\,{\it a2}\,{\it b2}+{\it b0} \right ) \sqrt {{{\it b2}}^{2}+2\,{\it b2}+4\,{\it c1}+1}-4\,{{\it a3}}^{2}{\it c1}+ \left ( \left ( {{\it b2}}^{2}+{\it b2}+4\,{\it c1} \right ) {\it a1}+ \left ( {{\it b2}}^{2}+{\it b2}+4\,{\it c1} \right ) {\it a2}+{\it b1}\, \left ( {\it b2}+1 \right ) \right ) {\it a3}- \left ( {{\it b2}}^{2}+{\it b2}+4\,{\it c1} \right ) {\it a2}\,{\it a1}+{\it b0}\, \left ( {\it b2}+1 \right ) \right ) \left ( 2\, \left ( {\it a2}-{\it a3} \right ) \left ( {\it a1}-{\it a3} \right ) \sqrt {{{\it b2}}^{2}+2\,{\it b2}+4\,{\it c1}+1}+ \left ( -2\,{\it b2}-2 \right ) {{\it a3}}^{2}+ \left ( 2\,{\it a1}+2\,{\it a2}-2\,{\it b1} \right ) {\it a3}-2\,{\it a1}\,{\it a2}-2\,{\it b0} \right ) ^{-1}},{\frac {-{\it b2}\,{{\it a1}}^{2}-{\it a1}\,{\it b1}-{\it b0}}{ \left ( {\it a1}-{\it a2} \right ) \left ( {\it a1}-{\it a3} \right ) }},{\frac {{\it b2}\,{{\it a2}}^{2}+{\it a2}\,{\it b1}+{\it b0}}{ \left ( {\it a1}-{\it a2} \right ) \left ( {\it a2}-{\it a3} \right ) }},{\frac {{\it a1}-x}{{\it a1}-{\it a2}}} \right ) +{\it \_C2}\,{\it HeunG} \left ( {\frac {{\it a1}-{\it a3}}{{\it a1}-{\it a2}}},{\frac {1}{ \left ( {\it a1}-{\it a2} \right ) ^{3} \left ( {\it a1}-{\it a3} \right ) ^{2}} \left ( - \left ( {\it a1}-{\it a2} \right ) \left ( {\it a1}\, \left ( {\it a1}-2\,{\it a2} \right ) {\it b2}-{\it a2}\,{\it b1}-{\it b0}+ \left ( -{\it a1}\,{\it c1}-{\it c0} \right ) {\it a2}+{{\it a1}}^{2}{\it c1}+{\it c0}\,{\it a1} \right ) {{\it a3}}^{2}+ \left ( {{\it a1}}^{3} \left ( {\it a1}-2\,{\it a2} \right ) {{\it b2}}^{2}+{\it a1}\, \left ( {\it a1}\, \left ( {\it a1}-3\,{\it a2} \right ) {\it b1}-2\,{\it a2}\,{\it b0}+{\it a1}\, \left ( {\it a1}-{\it a2} \right ) \left ( {\it a1}-3\,{\it a2} \right ) \right ) {\it b2}-{\it a1}\,{\it a2}\,{{\it b1}}^{2}- \left ( {\it a1}+{\it a2} \right ) \left ( {\it b0}+{\it a1}\, \left ( {\it a1}-{\it a2} \right ) \right ) {\it b1}-{{\it b0}}^{2}+ \left ( -3\,{{\it a1}}^{2}+4\,{\it a1}\,{\it a2}-{{\it a2}}^{2} \right ) {\it b0}+2\,{\it a1}\, \left ( {\it a1}-{\it a2} \right ) ^{2} \left ( {\it a1}\,{\it c1}+{\it c0} \right ) \right ) {\it a3}+{{\it a1}}^{4}{\it a2}\,{{\it b2}}^{2}+{{\it a1}}^{3} \left ( \left ( {\it a1}+{\it a2} \right ) {\it b1}+{\it a1}\,{\it a2}-{{\it a2}}^{2}+2\,{\it b0} \right ) {\it b2}+{{\it a1}}^{3}{{\it b1}}^{2}+{\it a1}\, \left ( \left ( 3\,{\it a1}-{\it a2} \right ) {\it b0}+{{\it a1}}^{2} \left ( {\it a1}-{\it a2} \right ) \right ) {\it b1}+ \left ( 2\,{\it a1}-{\it a2} \right ) {{\it b0}}^{2}+2\, \left ( {\it a1}-{\it a2} \right ) {\it a1}\, \left ( {\it a1}-{\it a2}/2 \right ) {\it b0}-{{\it a1}}^{2} \left ( {\it a1}-{\it a2} \right ) ^{2} \left ( {\it a1}\,{\it c1}+{\it c0} \right ) \right ) },{\frac {1}{ \left ( 2\,{\it a1}-2\,{\it a3} \right ) \left ( {\it a1}-{\it a2} \right ) } \left ( \left ( {\it a1}-{\it a3} \right ) \left ( {\it a1}-{\it a2} \right ) \sqrt {{{\it b2}}^{2}+2\,{\it b2}+4\,{\it c1}+1}+ \left ( {\it b2}+1 \right ) {{\it a1}}^{2}+ \left ( \left ( {\it b2}-1 \right ) {\it a2}+{\it a3}\, \left ( {\it b2}-1 \right ) +2\,{\it b1} \right ) {\it a1}-{\it a3}\, \left ( {\it b2}-1 \right ) {\it a2}+2\,{\it b0} \right ) },{\frac {1}{ \left ( 2\,{\it a1}-2\,{\it a2} \right ) \left ( {\it a1}-{\it a3} \right ) } \left ( \left ( \left ( {\it b2}+2 \right ) \left ( {\it a2}-{\it a3} \right ) {{\it a1}}^{2}+ \left ( 2\,{{\it a3}}^{2}-{\it b1}\,{\it a3}+ \left ( {\it b2}-2 \right ) {{\it a2}}^{2}+2\,{\it a2}\,{\it b1}+{\it b0} \right ) {\it a1}-2\,{\it a2}\,{{\it a3}}^{2}+ \left ( \left ( -{\it b2}+2 \right ) {{\it a2}}^{2}-{\it a2}\,{\it b1}-2\,{\it b0} \right ) {\it a3}+{\it a2}\,{\it b0} \right ) \left ( {\it a1}-{\it a3} \right ) \sqrt {{{\it b2}}^{2}+2\,{\it b2}+4\,{\it c1}+1}- \left ( {{\it b2}}^{2}+3\,{\it b2}+4\,{\it c1}+2 \right ) \left ( {\it a2}-{\it a3} \right ) {{\it a1}}^{3}+ \left ( \left ( -3\,{{\it b2}}^{2}-5\,{\it b2}-8\,{\it c1}-4 \right ) {{\it a3}}^{2}+ \left ( \left ( {{\it b2}}^{2}+3\,{\it b2}+4\,{\it c1}+2 \right ) {\it a2}- \left ( {\it b2}-1 \right ) {\it b1} \right ) {\it a3}+ \left ( {{\it b2}}^{2}+{\it b2}+4\,{\it c1}+2 \right ) {{\it a2}}^{2}-2\,{\it a2}\,{\it b1}-{\it b0}\, \left ( {\it b2}+1 \right ) \right ) {{\it a1}}^{2}+ \left ( \left ( 2\,{\it b2}+4\,{\it c1}+2 \right ) {{\it a3}}^{3}+ \left ( \left ( 2\,{\it b2}+4\,{\it c1}+2 \right ) {\it a2}-3\,{\it b2}\,{\it b1}-{\it b1} \right ) {{\it a3}}^{2}+ \left ( \left ( -2\,{{\it b2}}^{2}-2\,{\it b2}-8\,{\it c1}-4 \right ) {{\it a2}}^{2}-{\it b1}\, \left ( {\it b2}-3 \right ) {\it a2}-{\it b2}\,{\it b0}-2\,{{\it b1}}^{2}+3\,{\it b0} \right ) {\it a3}-{\it b0}\, \left ( \left ( {\it b2}+1 \right ) {\it a2}+2\,{\it b1} \right ) \right ) {\it a1}-2\,{\it a2}\, \left ( {\it b2}+2\,{\it c1}+1 \right ) {{\it a3}}^{3}+ \left ( \left ( {{\it b2}}^{2}+{\it b2}+4\,{\it c1}+2 \right ) {{\it a2}}^{2}+ \left ( {\it b2}-1 \right ) {\it b1}\,{\it a2}-2\,{\it b0}\, \left ( {\it b2}+1 \right ) \right ) {{\it a3}}^{2}+{\it b0}\, \left ( \left ( {\it b2}+1 \right ) {\it a2}-2\,{\it b1} \right ) {\it a3}-2\,{{\it b0}}^{2} \right ) \left ( \left ( {\it a2}-{\it a3} \right ) \left ( {\it a1}-{\it a3} \right ) \sqrt {{{\it b2}}^{2}+2\,{\it b2}+4\,{\it c1}+1}+ \left ( -{\it b2}-1 \right ) {{\it a3}}^{2}+ \left ( {\it a1}+{\it a2}-{\it b1} \right ) {\it a3}-{\it a1}\,{\it a2}-{\it b0} \right ) ^{-1}},{\frac { \left ( {\it b2}+2 \right ) {{\it a1}}^{2}+ \left ( -2\,{\it a2}-2\,{\it a3}+{\it b1} \right ) {\it a1}+2\,{\it a2}\,{\it a3}+{\it b0}}{ \left ( {\it a1}-{\it a2} \right ) \left ( {\it a1}-{\it a3} \right ) }},{\frac {{\it b2}\,{{\it a2}}^{2}+{\it a2}\,{\it b1}+{\it b0}}{ \left ( {\it a1}-{\it a2} \right ) \left ( {\it a2}-{\it a3} \right ) }},{\frac {{\it a1}-x}{{\it a1}-{\it a2}}} \right ) \left ( -{\it a1}+x \right ) ^{{\frac { \left ( {\it b2}+1 \right ) {{\it a1}}^{2}+ \left ( -{\it a2}-{\it a3}+{\it b1} \right ) {\it a1}+{\it a2}\,{\it a3}+{\it b0}}{ \left ( {\it a1}-{\it a2} \right ) \left ( {\it a1}-{\it a3} \right ) }}} \right \} \] Mathematica raw input
DSolve[(c0 + c1*x)*y[x] + (b0 + b1*x + b2*x^2)*y'[x] + (a1 - x)*(a2 - x)*(a3 - x)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {(-c0 - \[FormalX]*c1)*\[FormalY][\[FormalX]] + (-b0 - \[FormalX]*b1 - \[FormalX]^2*b2)*Derivative[1][\[FormalY]][\[FormalX]] - (-\[FormalX] + a1)*(-\[FormalX] + a2)*(-\[FormalX] + a3)*Derivative[2][\[FormalY]][\[FormalX]] == 0, \[FormalY][0] == C[1], Derivative[1][\[FormalY]][0] == C[2]}]][x]}}
Maple raw input
dsolve((a1-x)*(a2-x)*(a3-x)*diff(diff(y(x),x),x)+(b2*x^2+b1*x+b0)*diff(y(x),x)+(c1*x+c0)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*HeunG((a1-a3)/(a1-a2),(-a1*c1-c0)/(a1-a2),-1/2*b2-1/2+1/2*(b2^2+2*b2+4*c1+1)^(1/2),(((a1*b2+a2*b2+b1)*a3-a1*a2*b2+b0)*(b2^2+2*b2+4*c1+1)^(1/2)-4*a3^2*c1+((b2^2+b2+4*c1)*a1+(b2^2+b2+4*c1)*a2+b1*(b2+1))*a3-(b2^2+b2+4*c1)*a2*a1+b0*(b2+1))/(2*(a2-a3)*(a1-a3)*(b2^2+2*b2+4*c1+1)^(1/2)+(-2*b2-2)*a3^2+(2*a1+2*a2-2*b1)*a3-2*a1*a2-2*b0),1/(a1-a3)/(a1-a2)*(-a1^2*b2-a1*b1-b0),(a2^2*b2+a2*b1+b0)/(a1-a2)/(a2-a3),(a1-x)/(a1-a2))+_C2*HeunG((a1-a3)/(a1-a2),(-(a1-a2)*(a1*(a1-2*a2)*b2-a2*b1-b0+(-a1*c1-c0)*a2+a1^2*c1+c0*a1)*a3^2+(a1^3*(a1-2*a2)*b2^2+a1*(a1*(a1-3*a2)*b1-2*a2*b0+a1*(a1-a2)*(a1-3*a2))*b2-a1*a2*b1^2-(a1+a2)*(b0+a1*(a1-a2))*b1-b0^2+(-3*a1^2+4*a1*a2-a2^2)*b0+2*a1*(a1-a2)^2*(a1*c1+c0))*a3+a1^4*a2*b2^2+a1^3*((a1+a2)*b1+a1*a2-a2^2+2*b0)*b2+a1^3*b1^2+a1*((3*a1-a2)*b0+a1^2*(a1-a2))*b1+(2*a1-a2)*b0^2+2*(a1-a2)*a1*(a1-1/2*a2)*b0-a1^2*(a1-a2)^2*(a1*c1+c0))/(a1-a2)^3/(a1-a3)^2,1/2*((a1-a3)*(a1-a2)*(b2^2+2*b2+4*c1+1)^(1/2)+(b2+1)*a1^2+((b2-1)*a2+a3*(b2-1)+2*b1)*a1-a3*(b2-1)*a2+2*b0)/(a1-a3)/(a1-a2),1/2*(((b2+2)*(a2-a3)*a1^2+(2*a3^2-b1*a3+(b2-2)*a2^2+2*a2*b1+b0)*a1-2*a2*a3^2+((-b2+2)*a2^2-a2*b1-2*b0)*a3+a2*b0)*(a1-a3)*(b2^2+2*b2+4*c1+1)^(1/2)-(b2^2+3*b2+4*c1+2)*(a2-a3)*a1^3+((-3*b2^2-5*b2-8*c1-4)*a3^2+((b2^2+3*b2+4*c1+2)*a2-(b2-1)*b1)*a3+(b2^2+b2+4*c1+2)*a2^2-2*a2*b1-b0*(b2+1))*a1^2+((2*b2+4*c1+2)*a3^3+((2*b2+4*c1+2)*a2-3*b2*b1-b1)*a3^2+((-2*b2^2-2*b2-8*c1-4)*a2^2-b1*(b2-3)*a2-b2*b0-2*b1^2+3*b0)*a3-b0*((b2+1)*a2+2*b1))*a1-2*a2*(b2+2*c1+1)*a3^3+((b2^2+b2+4*c1+2)*a2^2+(b2-1)*b1*a2-2*b0*(b2+1))*a3^2+b0*((b2+1)*a2-2*b1)*a3-2*b0^2)/(a1-a2)/(a1-a3)/((a2-a3)*(a1-a3)*(b2^2+2*b2+4*c1+1)^(1/2)+(-b2-1)*a3^2+(a1+a2-b1)*a3-a1*a2-b0),((b2+2)*a1^2+(-2*a2-2*a3+b1)*a1+2*a2*a3+b0)/(a1-a3)/(a1-a2),(a2^2*b2+a2*b1+b0)/(a1-a2)/(a2-a3),(a1-x)/(a1-a2))*(-a1+x)^(((b2+1)*a1^2+(-a2-a3+b1)*a1+a2*a3+b0)/(a1-a3)/(a1-a2))