Internal problem ID [9860]
Internal file name [OUTPUT/8805_Monday_June_06_2022_05_31_20_AM_99333266/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1536.
ODE order: 4.
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
[[_high_order, _missing_x]]
\[ \boxed {y^{\prime \prime \prime \prime }+\lambda y=0} \] The characteristic equation is \[ \lambda ^{4}+\lambda = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= \left (-\lambda \right )^{\frac {1}{4}}\\ \lambda _2 &= i \left (-\lambda \right )^{\frac {1}{4}}\\ \lambda _3 &= -\left (-\lambda \right )^{\frac {1}{4}}\\ \lambda _4 &= -i \left (-\lambda \right )^{\frac {1}{4}} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{-i \left (-\lambda \right )^{\frac {1}{4}} x} c_{1} +{\mathrm e}^{\left (-\lambda \right )^{\frac {1}{4}} x} c_{2} +{\mathrm e}^{-\left (-\lambda \right )^{\frac {1}{4}} x} c_{3} +{\mathrm e}^{i \left (-\lambda \right )^{\frac {1}{4}} x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-i \left (-\lambda \right )^{\frac {1}{4}} x}\\ y_2 &= {\mathrm e}^{\left (-\lambda \right )^{\frac {1}{4}} x}\\ y_3 &= {\mathrm e}^{-\left (-\lambda \right )^{\frac {1}{4}} x}\\ y_4 &= {\mathrm e}^{i \left (-\lambda \right )^{\frac {1}{4}} x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-i \left (-\lambda \right )^{\frac {1}{4}} x} c_{1} +{\mathrm e}^{\left (-\lambda \right )^{\frac {1}{4}} x} c_{2} +{\mathrm e}^{-\left (-\lambda \right )^{\frac {1}{4}} x} c_{3} +{\mathrm e}^{i \left (-\lambda \right )^{\frac {1}{4}} x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{-i \left (-\lambda \right )^{\frac {1}{4}} x} c_{1} +{\mathrm e}^{\left (-\lambda \right )^{\frac {1}{4}} x} c_{2} +{\mathrm e}^{-\left (-\lambda \right )^{\frac {1}{4}} x} c_{3} +{\mathrm e}^{i \left (-\lambda \right )^{\frac {1}{4}} x} c_{4} \] Verified OK.
Maple trace
`Methods for high 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: 50
dsolve(diff(diff(diff(diff(y(x),x),x),x),x)+lambda*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-i \left (-\lambda \right )^{\frac {1}{4}} x}+c_{2} {\mathrm e}^{i \left (-\lambda \right )^{\frac {1}{4}} x}+c_{3} {\mathrm e}^{-\left (-\lambda \right )^{\frac {1}{4}} x}+c_{4} {\mathrm e}^{\left (-\lambda \right )^{\frac {1}{4}} x} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 76
DSolve[\[Lambda]*y[x] + Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 e^{(-1)^{3/4} \sqrt [4]{\lambda } x}+c_2 e^{-\sqrt [4]{-1} \sqrt [4]{\lambda } x}+c_3 e^{-(-1)^{3/4} \sqrt [4]{\lambda } x}+c_4 e^{\sqrt [4]{-1} \sqrt [4]{\lambda } x} \]