4.34.47 \((1-x) x (a-x) y''(x)+y'(x) \left (x^2 (\text {a0}+\text {a1}+1)+\text {a0} \text {a2}-\text {a0}+\text {a1}+x (\text {a2}+\text {a3})-\text {a3}+1\right )+\text {a0} \text {a1} (x-k) y(x)=0\)

ODE
\[ (1-x) x (a-x) y''(x)+y'(x) \left (x^2 (\text {a0}+\text {a1}+1)+\text {a0} \text {a2}-\text {a0}+\text {a1}+x (\text {a2}+\text {a3})-\text {a3}+1\right )+\text {a0} \text {a1} (x-k) y(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 9.51155 (sec), leaf count = 0 , DifferentialRoot result

\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{-\text {a0} \text {a1} (k-\unicode {f817}) \unicode {f818}(\unicode {f817})+\left (\text {a0} \unicode {f817}^2+\text {a1} \unicode {f817}^2+\unicode {f817}^2+\text {a2} \unicode {f817}+\text {a3} \unicode {f817}-\text {a0}+\text {a1}+\text {a0} \text {a2}-\text {a3}+1\right ) \unicode {f818}'(\unicode {f817})-(\unicode {f817}-1) \unicode {f817} (a-\unicode {f817}) \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(2)=c_1,\unicode {f818}'(2)=c_2\right \}\right )(x)\right \}\right \}\]

Maple ✓
cpu = 0.778 (sec), leaf count = 393

\[ \left \{ y \left ( x \right ) = \left ( -1+x \right ) ^{{\frac {{\it a0}\,{\it a2}+a+2\,{\it a1}+{\it a2}+1}{a-1}}} \left ( {\it HeunG} \left ( a,{\frac {2\,{a}^{4}+ \left ( {\it a0}\,{\it a1}\,k+2\,{\it a0}+2\,{\it a1}+{\it a2}+{\it a3} \right ) {a}^{3}+ \left ( \left ( 1-{\it a2} \right ) {{\it a0}}^{2}+ \left ( \left ( -k-{\it a2} \right ) {\it a1}+{\it a3}-1 \right ) {\it a0}-{{\it a1}}^{2}+ \left ( {\it a3}-1 \right ) {\it a1}+{\it a2}+{\it a3} \right ) {a}^{2}- \left ( {\it a2}+{\it a3}-1 \right ) \left ( \left ( -1+{\it a2} \right ) {\it a0}+{\it a1}-{\it a3}+1 \right ) a- \left ( \left ( -1+{\it a2} \right ) {\it a0}+{\it a1}-{\it a3}+1 \right ) ^{2}}{{a}^{2} \left ( a-1 \right ) }},{\frac { \left ( {\it a0}+2 \right ) {a}^{2}+ \left ( {\it a1}+{\it a2}+{\it a3}-1 \right ) a+ \left ( -1+{\it a2} \right ) {\it a0}+{\it a1}-{\it a3}+1}{a \left ( a-1 \right ) }},{\frac { \left ( {\it a1}+2 \right ) {a}^{2}+ \left ( {\it a0}+{\it a2}+{\it a3}-1 \right ) a+ \left ( -1+{\it a2} \right ) {\it a0}+{\it a1}-{\it a3}+1}{a \left ( a-1 \right ) }},{\frac {-{\it a0}\,{\it a2}+2\,a+{\it a0}-{\it a1}+{\it a3}-1}{a}},{\frac {{\it a0}\,{\it a2}+2\,a+2\,{\it a1}+{\it a2}}{a-1}},x \right ) {x}^{{\frac {-{\it a0}\,{\it a2}+a+{\it a0}-{\it a1}+{\it a3}-1}{a}}}{\it \_C2}+{\it HeunG} \left ( a,{\frac { \left ( {{\it a2}}^{2}-{\it a2} \right ) {{\it a0}}^{2}+ \left ( {{\it a2}}^{2}+ \left ( a+3\,{\it a1}-{\it a3}+1 \right ) {\it a2}+ \left ( ka-k-2 \right ) {\it a1}-a-1 \right ) {\it a0}+ \left ( {\it a1}-{\it a3}+1 \right ) \left ( a+2\,{\it a1}+{\it a2}+1 \right ) }{a-1}},{\frac { \left ( a+{\it a2}-1 \right ) {\it a0}+a+2\,{\it a1}+{\it a2}+1}{a-1}},{\frac {a{\it a1}+{\it a0}\,{\it a2}+a+{\it a1}+{\it a2}+1}{a-1}},{\frac { \left ( -1+{\it a2} \right ) {\it a0}+{\it a1}-{\it a3}+1}{a}},{\frac {{\it a0}\,{\it a2}+2\,a+2\,{\it a1}+{\it a2}}{a-1}},x \right ) {\it \_C1} \right ) \right \} \] 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] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {-(a0*a1*(-\[Forma
lX] + k)*\[FormalY][\[FormalX]]) + (1 + \[FormalX]^2 - a0 + \[FormalX]^2*a0 + a1
 + \[FormalX]^2*a1 + \[FormalX]*a2 + a0*a2 - a3 + \[FormalX]*a3)*Derivative[1][\
[FormalY]][\[FormalX]] - (-1 + \[FormalX])*\[FormalX]*(-\[FormalX] + a)*Derivati
ve[2][\[FormalY]][\[FormalX]] == 0, \[FormalY][2] == C[1], Derivative[1][\[Forma
lY]][2] == C[2]}]][x]}}

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),'implicit')

Maple raw output

y(x) = (-1+x)^((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+((1-a2)*a0^2+((-k-a2)*a1+a3-1)*a0-a1^2+(a3-1)*a1+a2+a3)*a^2-(a2+a3-1)*((
-1+a2)*a0+a1-a3+1)*a-((-1+a2)*a0+a1-a3+1)^2)/a^2/(a-1),((a0+2)*a^2+(a1+a2+a3-1)*
a+(-1+a2)*a0+a1-a3+1)/a/(a-1),((a1+2)*a^2+(a0+a2+a3-1)*a+(-1+a2)*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)*x^((-a0*a2+a+a0-a1+a
3-1)/a)*_C2+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)*(a+2*a1+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),((-1+a2)*a0+a1-a3+1)/a,(a0*a2+2*a+2*a1+a2)/(a-1),x)*_C1)