Internal problem ID [9106]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1527.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x -a \right )^{3} \left (x -b \right )^{3} y^{\prime \prime \prime }-c y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.021 (sec). Leaf size: 500
dsolve((x-a)^3*(x-b)^3*diff(diff(diff(y(x),x),x),x)-c*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (-b +x \right )^{\frac {2 a}{a -b}} \left (x -a \right )^{-\frac {2 b}{a -b}} \left (b -x \right )^{-\frac {\RootOf \left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \mathit {index} =1\right )}{a -b}} \left (a -x \right )^{\frac {\RootOf \left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \mathit {index} =1\right )}{a -b}}+c_{2} \left (-b +x \right )^{\frac {2 a}{a -b}} \left (x -a \right )^{-\frac {2 b}{a -b}} \left (b -x \right )^{-\frac {\RootOf \left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \mathit {index} =2\right )}{a -b}} \left (a -x \right )^{\frac {\RootOf \left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \mathit {index} =2\right )}{a -b}}+c_{3} \left (-b +x \right )^{\frac {2 a}{a -b}} \left (x -a \right )^{-\frac {2 b}{a -b}} \left (b -x \right )^{-\frac {\RootOf \left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \mathit {index} =3\right )}{a -b}} \left (a -x \right )^{\frac {\RootOf \left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \mathit {index} =3\right )}{a -b}} \]
✓ Solution by Mathematica
Time used: 130.086 (sec). Leaf size: 152
DSolve[-(c*y[x]) + (-a + x)^3*(-b + x)^3*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} 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 ) \\ \end{align*}