Internal problem ID [9849]
Internal file name [OUTPUT/8794_Monday_June_06_2022_05_30_01_AM_73498078/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1525.
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.
\[ \boxed {x^{6} y^{\prime \prime \prime }+6 x^{5} y^{\prime \prime }+a y=0} \] 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: 0.032 (sec). Leaf size: 212
dsolve(x^6*diff(diff(diff(y(x),x),x),x)+6*x^5*diff(diff(y(x),x),x)+a*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {4 \left (-8 x^{3}+a \right )^{4} \left (c_{3} {\mathrm e}^{\frac {\left (-a^{4}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2 a x}} \left (\frac {\left (-i+\sqrt {3}\right ) \left (-a^{4}\right )^{\frac {1}{3}}}{4}+i a x \right )+\left (\frac {\left (\sqrt {3}+i\right ) \left (-a^{4}\right )^{\frac {1}{3}}}{4}-i a x \right ) c_{2} {\mathrm e}^{-\frac {\left (-a^{4}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{2 a x}}+128 \,{\mathrm e}^{-\frac {\left (-a^{4}\right )^{\frac {1}{3}}}{a x}} c_{1} \left (a x +\frac {\left (-a^{4}\right )^{\frac {1}{3}}}{2}\right )\right )}{{\left (2 a x +\left (-a^{4}\right )^{\frac {1}{3}}\right )}^{4} {\left (i \left (-a^{4}\right )^{\frac {1}{3}} \sqrt {3}-4 a x +\left (-a^{4}\right )^{\frac {1}{3}}\right )}^{4} {\left (i \left (-a^{4}\right )^{\frac {1}{3}} \sqrt {3}+4 a x -\left (-a^{4}\right )^{\frac {1}{3}}\right )}^{4}} \]
✓ Solution by Mathematica
Time used: 0.366 (sec). Leaf size: 101
DSolve[a*y[x] + 6*x^5*y''[x] + x^6*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \left (-e^{\frac {\sqrt [3]{a}}{x}}\right ) \left (\sqrt [3]{a}-2 x\right )+c_2 e^{\frac {(-1)^{2/3} \sqrt [3]{a}}{x}} \left (x-\frac {1}{2} (-1)^{2/3} \sqrt [3]{a}\right )+c_3 e^{-\frac {\sqrt [3]{-1} \sqrt [3]{a}}{x}} \left (\frac {1}{2} \sqrt [3]{-1} \sqrt [3]{a}+x\right ) \]