##### 4.45.9 $$(a-x)^3 (b-x)^3 y'''(x)=c y(x)$$

ODE
$(a-x)^3 (b-x)^3 y'''(x)=c y(x)$ ODE Classiﬁcation

[[_3rd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica
cpu = 130.294 (sec), leaf count = 152

$\left \{\left \{y(x)\to (b-x)^2 \left (c_1 \left (\frac {a-x}{b-x}\right )^{\text {Root}\left [-\text {\#1}^3+3 \text {\#1}^2-2 \text {\#1}+\frac {c}{(a-b)^3}\& ,1\right ]}+c_2 \left (\frac {a-x}{b-x}\right )^{\text {Root}\left [-\text {\#1}^3+3 \text {\#1}^2-2 \text {\#1}+\frac {c}{(a-b)^3}\& ,2\right ]}+c_3 \left (\frac {a-x}{b-x}\right )^{\text {Root}\left [-\text {\#1}^3+3 \text {\#1}^2-2 \text {\#1}+\frac {c}{(a-b)^3}\& ,3\right ]}\right )\right \}\right \}$

Maple
cpu = 0.956 (sec), leaf count = 500

$\left [y \left (x \right ) = \textit {\_C1} \left (x -a \right )^{-\frac {2 b}{a -b}} \left (x -b \right )^{\frac {2 a}{a -b}} \left (b -x \right )^{-\frac {\RootOf \left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \mathit {index} =1\right )}{a -b}} \left (a -x \right )^{\frac {\RootOf \left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \mathit {index} =1\right )}{a -b}}+\textit {\_C2} \left (x -a \right )^{-\frac {2 b}{a -b}} \left (x -b \right )^{\frac {2 a}{a -b}} \left (b -x \right )^{-\frac {\RootOf \left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \mathit {index} =2\right )}{a -b}} \left (a -x \right )^{\frac {\RootOf \left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \mathit {index} =2\right )}{a -b}}+\textit {\_C3} \left (x -a \right )^{-\frac {2 b}{a -b}} \left (x -b \right )^{\frac {2 a}{a -b}} \left (b -x \right )^{-\frac {\RootOf \left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \mathit {index} =3\right )}{a -b}} \left (a -x \right )^{\frac {\RootOf \left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \mathit {index} =3\right )}{a -b}}\right ]$ Mathematica raw input

DSolve[(a - x)^3*(b - x)^3*y'''[x] == c*y[x],y[x],x]

Mathematica raw output

{{y[x] -> (b - x)^2*(((a - x)/(b - x))^Root[c/(a - b)^3 - 2*#1 + 3*#1^2 - #1^3 &
 , 1]*C[1] + ((a - x)/(b - x))^Root[c/(a - b)^3 - 2*#1 + 3*#1^2 - #1^3 & , 2]*C[
2] + ((a - x)/(b - x))^Root[c/(a - b)^3 - 2*#1 + 3*#1^2 - #1^3 & , 3]*C[3])}}

Maple raw input

dsolve((a-x)^3*(b-x)^3*diff(diff(diff(y(x),x),x),x) = c*y(x), y(x))

Maple raw output

[y(x) = _C1*(x-a)^(-2/(a-b)*b)*(x-b)^(2*a/(a-b))*(b-x)^(-1/(a-b)*RootOf(_Z^3+(-3
*a-3*b)*_Z^2+(2*a^2+8*a*b+2*b^2)*_Z-4*a^2*b-4*a*b^2-c,index = 1))*(a-x)^(1/(a-b)
*RootOf(_Z^3+(-3*a-3*b)*_Z^2+(2*a^2+8*a*b+2*b^2)*_Z-4*a^2*b-4*a*b^2-c,index = 1)
)+_C2*(x-a)^(-2/(a-b)*b)*(x-b)^(2*a/(a-b))*(b-x)^(-1/(a-b)*RootOf(_Z^3+(-3*a-3*b
)*_Z^2+(2*a^2+8*a*b+2*b^2)*_Z-4*a^2*b-4*a*b^2-c,index = 2))*(a-x)^(1/(a-b)*RootO
f(_Z^3+(-3*a-3*b)*_Z^2+(2*a^2+8*a*b+2*b^2)*_Z-4*a^2*b-4*a*b^2-c,index = 2))+_C3*
(x-a)^(-2/(a-b)*b)*(x-b)^(2*a/(a-b))*(b-x)^(-1/(a-b)*RootOf(_Z^3+(-3*a-3*b)*_Z^2
+(2*a^2+8*a*b+2*b^2)*_Z-4*a^2*b-4*a*b^2-c,index = 3))*(a-x)^(1/(a-b)*RootOf(_Z^3
+(-3*a-3*b)*_Z^2+(2*a^2+8*a*b+2*b^2)*_Z-4*a^2*b-4*a*b^2-c,index = 3))]