Internal problem ID [9775]
Internal file name [OUTPUT/8718_Monday_June_06_2022_05_19_41_AM_93213124/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1449.
ODE order: 3.
ODE degree: 1.
The type(s) of ODE detected by this program : "higher_order_linear_constant_coefficients_ODE"
Maple gives the following as the ode type
[[_3rd_order, _missing_x]]
\[ \boxed {y^{\prime \prime \prime }-\lambda y=0} \] The characteristic equation is \[ \lambda ^{3}-\lambda = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= \lambda ^{\frac {1}{3}}\\ \lambda _2 &= -\frac {\lambda ^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \lambda ^{\frac {1}{3}}}{2}\\ \lambda _3 &= -\frac {\lambda ^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \lambda ^{\frac {1}{3}}}{2} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (-\frac {\lambda ^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \lambda ^{\frac {1}{3}}}{2}\right ) x} c_{1} +{\mathrm e}^{\lambda ^{\frac {1}{3}} x} c_{2} +{\mathrm e}^{\left (-\frac {\lambda ^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \lambda ^{\frac {1}{3}}}{2}\right ) x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{\left (-\frac {\lambda ^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \lambda ^{\frac {1}{3}}}{2}\right ) x}\\ y_2 &= {\mathrm e}^{\lambda ^{\frac {1}{3}} x}\\ y_3 &= {\mathrm e}^{\left (-\frac {\lambda ^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \lambda ^{\frac {1}{3}}}{2}\right ) x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\left (-\frac {\lambda ^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \lambda ^{\frac {1}{3}}}{2}\right ) x} c_{1} +{\mathrm e}^{\lambda ^{\frac {1}{3}} x} c_{2} +{\mathrm e}^{\left (-\frac {\lambda ^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \lambda ^{\frac {1}{3}}}{2}\right ) x} c_{3} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{\left (-\frac {\lambda ^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \lambda ^{\frac {1}{3}}}{2}\right ) x} c_{1} +{\mathrm e}^{\lambda ^{\frac {1}{3}} x} c_{2} +{\mathrm e}^{\left (-\frac {\lambda ^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \lambda ^{\frac {1}{3}}}{2}\right ) x} c_{3} \] Verified OK.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 47
dsolve(diff(diff(diff(y(x),x),x),x)-lambda*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {\lambda ^{\frac {1}{3}} \left (1+i \sqrt {3}\right ) x}{2}}+c_{2} {\mathrm e}^{\frac {\lambda ^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) x}{2}}+c_{3} {\mathrm e}^{\lambda ^{\frac {1}{3}} x} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 53
DSolve[-(\[Lambda]*y[x]) + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 e^{(-1)^{2/3} \sqrt [3]{\lambda } x}+c_2 e^{-\sqrt [3]{-1} \sqrt [3]{\lambda } x}+c_3 e^{\sqrt [3]{\lambda } x} \]