Internal problem ID [5985]
Internal file name [OUTPUT/5233_Sunday_June_05_2022_03_28_05_PM_69631145/index.tex]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 2. Linear equations with constant coefficients. Page 83
Problem number: 1(d).
ODE order: 5.
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]]
\[ \boxed {y^{\left (5\right )}+2 y=0} \] The characteristic equation is \[ \lambda ^{5}+2 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= \cos \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}+i \sin \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}\\ \lambda _2 &= \left (\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )-\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\\ \lambda _3 &= \left (\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )-\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\\ \lambda _4 &= \left (\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )+\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (-\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\\ \lambda _5 &= \left (\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )+\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (-\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (\left (\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )-\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\right ) x} c_{1} +{\mathrm e}^{\left (\left (\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )+\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (-\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\right ) x} c_{2} +{\mathrm e}^{\left (\cos \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}+i \sin \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}\right ) x} c_{3} +{\mathrm e}^{\left (\left (\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )-\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\right ) x} c_{4} +{\mathrm e}^{\left (\left (\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )+\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (-\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\right ) x} c_{5} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{\left (\left (\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )-\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\right ) x}\\ y_2 &= {\mathrm e}^{\left (\left (\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )+\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (-\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\right ) x}\\ y_3 &= {\mathrm e}^{\left (\cos \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}+i \sin \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}\right ) x}\\ y_4 &= {\mathrm e}^{\left (\left (\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )-\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\right ) x}\\ y_5 &= {\mathrm e}^{\left (\left (\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )+\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (-\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\right ) x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\left (\left (\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )-\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\right ) x} c_{1} +{\mathrm e}^{\left (\left (\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )+\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (-\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\right ) x} c_{2} +{\mathrm e}^{\left (\cos \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}+i \sin \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}\right ) x} c_{3} +{\mathrm e}^{\left (\left (\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )-\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\right ) x} c_{4} +{\mathrm e}^{\left (\left (\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )+\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (-\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\right ) x} c_{5} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{\left (\left (\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )-\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\right ) x} c_{1} +{\mathrm e}^{\left (\left (\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )+\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (-\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\right ) x} c_{2} +{\mathrm e}^{\left (\cos \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}+i \sin \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}\right ) x} c_{3} +{\mathrm e}^{\left (\left (\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )-\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\right ) x} c_{4} +{\mathrm e}^{\left (\left (\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \cos \left (\frac {\pi }{5}\right )+\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}\right ) 2^{\frac {1}{5}}+i \left (-\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\left (\frac {\sqrt {5}}{4}-\frac {1}{4}\right ) \sin \left (\frac {\pi }{5}\right )\right ) 2^{\frac {1}{5}}\right ) x} c_{5} \] 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: 0.0 (sec). Leaf size: 140
dsolve(diff(y(x),x$5)+2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {\left (-i 2^{\frac {7}{10}} \sqrt {5-\sqrt {5}}+2^{\frac {1}{5}} \sqrt {5}+2^{\frac {1}{5}}\right ) x}{4}}+c_{2} {\mathrm e}^{-\frac {x \left (i \left (\sqrt {5}+1\right ) 2^{\frac {7}{10}} \sqrt {5-\sqrt {5}}+2 \,2^{\frac {1}{5}} \left (\sqrt {5}-1\right )\right )}{8}}+c_{3} {\mathrm e}^{-2^{\frac {1}{5}} x}+c_{4} {\mathrm e}^{\frac {\left (i \left (\sqrt {5}+1\right ) 2^{\frac {7}{10}} \sqrt {5-\sqrt {5}}-2 \,2^{\frac {1}{5}} \left (\sqrt {5}-1\right )\right ) x}{8}}+c_{5} {\mathrm e}^{2^{\frac {1}{5}} \left (\cos \left (\frac {\pi }{5}\right )+i \sin \left (\frac {\pi }{5}\right )\right ) x} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 180
DSolve[y'''''[x]+2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {\left (\sqrt {5}-1\right ) x}{2\ 2^{4/5}}} \left (c_5 e^{\frac {\left (\sqrt {5}-5\right ) x}{2\ 2^{4/5}}}+c_3 e^{\frac {\sqrt {5} x}{2^{4/5}}} \cos \left (\frac {\sqrt {5-\sqrt {5}} x}{2\ 2^{3/10}}\right )+c_4 \cos \left (\frac {\sqrt {5+\sqrt {5}} x}{2\ 2^{3/10}}\right )+c_2 e^{\frac {\sqrt {5} x}{2^{4/5}}} \sin \left (\frac {\sqrt {5-\sqrt {5}} x}{2\ 2^{3/10}}\right )+c_1 \sin \left (\frac {\sqrt {5+\sqrt {5}} x}{2\ 2^{3/10}}\right )\right ) \]