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

\[ \left \{ y \left ( x \right ) = \left ( x-b \right ) ^{2\,{\frac {a}{a-b}}} \left ( x-a \right ) ^{-2\,{\frac {b}{a-b}}} \left ( \left ( b-x \right ) ^{-{\frac {{\it RootOf} \left ( {{\it \_Z}}^{3}+ \left ( -3\,a-3\,b \right ) {{\it \_Z}}^{2}+ \left ( 2\,{a}^{2}+8\,ab+2\,{b}^{2} \right ) {\it \_Z}-4\,{a}^{2}b-4\,a{b}^{2}-c,{\it index}=3 \right ) }{a-b}}} \left ( a-x \right ) ^{{\frac {{\it RootOf} \left ( {{\it \_Z}}^{3}+ \left ( -3\,a-3\,b \right ) {{\it \_Z}}^{2}+ \left ( 2\,{a}^{2}+8\,ab+2\,{b}^{2} \right ) {\it \_Z}-4\,{a}^{2}b-4\,a{b}^{2}-c,{\it index}=3 \right ) }{a-b}}}{\it \_C3}+ \left ( b-x \right ) ^{-{\frac {{\it RootOf} \left ( {{\it \_Z}}^{3}+ \left ( -3\,a-3\,b \right ) {{\it \_Z}}^{2}+ \left ( 2\,{a}^{2}+8\,ab+2\,{b}^{2} \right ) {\it \_Z}-4\,{a}^{2}b-4\,a{b}^{2}-c,{\it index}=2 \right ) }{a-b}}} \left ( a-x \right ) ^{{\frac {{\it RootOf} \left ( {{\it \_Z}}^{3}+ \left ( -3\,a-3\,b \right ) {{\it \_Z}}^{2}+ \left ( 2\,{a}^{2}+8\,ab+2\,{b}^{2} \right ) {\it \_Z}-4\,{a}^{2}b-4\,a{b}^{2}-c,{\it index}=2 \right ) }{a-b}}}{\it \_C2}+ \left ( b-x \right ) ^{-{\frac {{\it RootOf} \left ( {{\it \_Z}}^{3}+ \left ( -3\,a-3\,b \right ) {{\it \_Z}}^{2}+ \left ( 2\,{a}^{2}+8\,ab+2\,{b}^{2} \right ) {\it \_Z}-4\,{a}^{2}b-4\,a{b}^{2}-c,{\it index}=1 \right ) }{a-b}}} \left ( a-x \right ) ^{{\frac {{\it RootOf} \left ( {{\it \_Z}}^{3}+ \left ( -3\,a-3\,b \right ) {{\it \_Z}}^{2}+ \left ( 2\,{a}^{2}+8\,ab+2\,{b}^{2} \right ) {\it \_Z}-4\,{a}^{2}b-4\,a{b}^{2}-c,{\it index}=1 \right ) }{a-b}}}{\it \_C1} \right ) \right \} \] 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)