Internal problem ID [5232]
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.
CAS Maple gives this as type [[_high_order, _missing_x]]
Solve \begin {gather*} \boxed {y^{\relax (5)}+2 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 369
dsolve(diff(y(x),x$5)+2*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{\left (\frac {2^{\frac {1}{5}} \cos \left (\frac {\pi }{5}\right ) \sqrt {5}}{4}-\frac {\cos \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}}{4}+\frac {2^{\frac {7}{10}} \sqrt {5+\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}-\frac {i 2^{\frac {7}{10}} \sqrt {5+\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\frac {i 2^{\frac {1}{5}} \sin \left (\frac {\pi }{5}\right ) \sqrt {5}}{4}-\frac {i \sin \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}}{4}\right ) x}+c_{2} {\mathrm e}^{\left (-\frac {2^{\frac {1}{5}} \cos \left (\frac {\pi }{5}\right ) \sqrt {5}}{4}-\frac {\cos \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}}{4}+\frac {2^{\frac {7}{10}} \sqrt {5-\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}-\frac {i 2^{\frac {7}{10}} \sqrt {5-\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}-\frac {i 2^{\frac {1}{5}} \sin \left (\frac {\pi }{5}\right ) \sqrt {5}}{4}-\frac {i \sin \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}}{4}\right ) x}+c_{3} {\mathrm e}^{\left (-\frac {2^{\frac {1}{5}} \cos \left (\frac {\pi }{5}\right ) \sqrt {5}}{4}-\frac {\cos \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}}{4}-\frac {2^{\frac {7}{10}} \sqrt {5-\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}+\frac {i 2^{\frac {7}{10}} \sqrt {5-\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}-\frac {i 2^{\frac {1}{5}} \sin \left (\frac {\pi }{5}\right ) \sqrt {5}}{4}-\frac {i \sin \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}}{4}\right ) x}+c_{4} {\mathrm e}^{\left (\frac {2^{\frac {1}{5}} \cos \left (\frac {\pi }{5}\right ) \sqrt {5}}{4}-\frac {\cos \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}}{4}-\frac {2^{\frac {7}{10}} \sqrt {5+\sqrt {5}}\, \sin \left (\frac {\pi }{5}\right )}{4}+\frac {i 2^{\frac {7}{10}} \sqrt {5+\sqrt {5}}\, \cos \left (\frac {\pi }{5}\right )}{4}+\frac {i 2^{\frac {1}{5}} \sin \left (\frac {\pi }{5}\right ) \sqrt {5}}{4}-\frac {i \sin \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}}}{4}\right ) x}+c_{5} {\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} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 168
DSolve[y'''''[x]+2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} 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_4 \cos \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 )+e^{\frac {\sqrt {5} x}{2^{4/5}}} \left (c_3 \cos \left (\frac {\sqrt {5-\sqrt {5}} x}{2\ 2^{3/10}}\right )+c_2 \sin \left (\frac {\sqrt {5-\sqrt {5}} x}{2\ 2^{3/10}}\right )\right )\right ) \\ \end{align*}