Internal problem ID [9851]
Internal file name [OUTPUT/8796_Monday_June_06_2022_05_30_15_AM_59637969/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_3rd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {\left (x -a \right )^{3} \left (x -b \right )^{3} y^{\prime \prime \prime }-c y=0} \] Unable to solve this ODE.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying high order exact linear fully integrable trying to convert to a linear ODE with constant coefficients trying differential order: 3; missing the dependent variable trying Louvillian solutions for 3rd order ODEs, imprimitive case`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 437
dsolve((x-a)^3*(x-b)^3*diff(diff(diff(y(x),x),x),x)-c*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (x -a \right )^{-\frac {2 b}{a -b}} \left (x -b \right )^{\frac {2 a}{a -b}} \left (c_{1} \left (-x +b \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =1\right )}{a -b}} \left (-x +a \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =1\right )}{a -b}}+c_{2} \left (-x +b \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =2\right )}{a -b}} \left (-x +a \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =2\right )}{a -b}}+c_{3} \left (-x +b \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =3\right )}{a -b}} \left (-x +a \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =3\right )}{a -b}}\right ) \]
✓ Solution by Mathematica
Time used: 130.138 (sec). Leaf size: 165
DSolve[-(c*y[x]) + (-a + x)^3*(-b + x)^3*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 (x-b)^2 \left (\frac {x-a}{x-b}\right )^{\text {Root}\left [-\text {$\#$1}^3+3 \text {$\#$1}^2-2 \text {$\#$1}+\frac {c}{(a-b)^3}\&,1\right ]}+c_2 (x-b)^2 \left (\frac {x-a}{x-b}\right )^{\text {Root}\left [-\text {$\#$1}^3+3 \text {$\#$1}^2-2 \text {$\#$1}+\frac {c}{(a-b)^3}\&,2\right ]}+c_3 (x-b)^2 \left (\frac {x-a}{x-b}\right )^{\text {Root}\left [-\text {$\#$1}^3+3 \text {$\#$1}^2-2 \text {$\#$1}+\frac {c}{(a-b)^3}\&,3\right ]} \]