(a −x)3(b −x)3y′′′(x) = cy(x)

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 = 135.011 (sec), leaf count = 0 , DiﬀerentialRoot result

$\left \{\left \{y(x)\to \text {DifferentialRoot}\left [\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{(a-\unicode {f817})^3 (b-\unicode {f817})^3 \unicode {f818}^{\text {Symbol}[\text {StringJoin}[\text {ConstantArray}[\prime ,3]]]}(\unicode {f817})-c \unicode {f818}(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2,\unicode {f818}''(0)=c_3\right \}\right ][x]\right \}\right \}$

Maple ✓
cpu = 0.437 (sec), leaf count = 437

${y (x) = {(x - b)}^{2 \frac{a}{a - b}} {(x - a)}^{- 2 \frac{b}{a - b}} ({(b - x)}^{- \frac{R o o t O f ({_Z}^{3} + (- 3 a - 3 b) {_Z}^{2} + (2 a^{2} + 8 a b + 2 b^{2})_Z - 4 a^{2} b - 4 a b^{2} - c, i n d e x = 3)}{a - b}} {(a - x)}^{\frac{R o o t O f ({_Z}^{3} + (- 3 a - 3 b) {_Z}^{2} + (2 a^{2} + 8 a b + 2 b^{2})_Z - 4 a^{2} b - 4 a b^{2} - c, i n d e x = 3)}{a - b}}_C 3 + {(b - x)}^{- \frac{R o o t O f ({_Z}^{3} + (- 3 a - 3 b) {_Z}^{2} + (2 a^{2} + 8 a b + 2 b^{2})_Z - 4 a^{2} b - 4 a b^{2} - c, i n d e x = 2)}{a - b}} {(a - x)}^{\frac{R o o t O f ({_Z}^{3} + (- 3 a - 3 b) {_Z}^{2} + (2 a^{2} + 8 a b + 2 b^{2})_Z - 4 a^{2} b - 4 a b^{2} - c, i n d e x = 2)}{a - b}}_C 2 + {(b - x)}^{- \frac{R o o t O f ({_Z}^{3} + (- 3 a - 3 b) {_Z}^{2} + (2 a^{2} + 8 a b + 2 b^{2})_Z - 4 a^{2} b - 4 a b^{2} - c, i n d e x = 1)}{a - b}} {(a - x)}^{\frac{R o o t O f ({_Z}^{3} + (- 3 a - 3 b) {_Z}^{2} + (2 a^{2} + 8 a b + 2 b^{2})_Z - 4 a^{2} b - 4 a b^{2} - c, i n d e x = 1)}{a - b}}_C 1)}$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {-(c*\[FormalY][\[
FormalX]]) + (-\[FormalX] + a)^3*(-\[FormalX] + b)^3*Derivative[3][\[FormalY]][\
[FormalX]] == 0, \[FormalY][0] == C[1], Derivative[1][\[FormalY]][0] == C[2], De
rivative[2][\[FormalY]][0] == C[3]}]][x]}}

Maple raw input

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

Maple raw output

y(x) = (x-b)^(2/(a-b)*a)*(x-a)^(-2/(a-b)*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)*Roo
tOf(_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))*_C
3+(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)*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))*_C2+(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))*_C1)

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4.45.9 (a−x)3(b−x)3y‴(x)=cy(x)

4.45.9 $(a - x)^{3} (b - x)^{3} y^{‴} (x) = c y (x)$