49.9.4 problem 1(d)

Internal problem ID [7653]
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)
Date solved : Monday, January 27, 2025 at 03:09:04 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\left (5\right )}+2 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 136

dsolve(diff(y(x),x$5)+2*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} {\mathrm e}^{-\frac {x \left (i 2^{{7}/{10}} \sqrt {5-\sqrt {5}}-2^{{1}/{5}} \left (\sqrt {5}+1\right )\right )}{4}}+c_{2} {\mathrm e}^{-\frac {x \left (i 2^{{7}/{10}} \left (\sqrt {5}+1\right ) \sqrt {5-\sqrt {5}}+2 \left (\sqrt {5}-1\right ) 2^{{1}/{5}}\right )}{8}}+c_3 \,{\mathrm e}^{-2^{{1}/{5}} x}+c_4 \,{\mathrm e}^{\frac {x \left (i 2^{{7}/{10}} \left (\sqrt {5}+1\right ) \sqrt {5-\sqrt {5}}-2 \left (\sqrt {5}-1\right ) 2^{{1}/{5}}\right )}{8}}+c_5 \,{\mathrm e}^{2^{{1}/{5}} \left (\cos \left (\frac {\pi }{5}\right )+i \sin \left (\frac {\pi }{5}\right )\right ) x} \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 180

DSolve[D[y[x],{x,5}]+2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-\frac {\left (\sqrt {5}-1\right ) x}{2\ 2^{4/5}}} \left (c_5 e^{\frac {\left (\sqrt {5}-5\right ) x}{2\ 2^{4/5}}}+c_3 e^{\frac {\sqrt {5} x}{2^{4/5}}} \cos \left (\frac {\sqrt {5-\sqrt {5}} x}{2\ 2^{3/10}}\right )+c_4 \cos \left (\frac {\sqrt {5+\sqrt {5}} x}{2\ 2^{3/10}}\right )+c_2 e^{\frac {\sqrt {5} x}{2^{4/5}}} \sin \left (\frac {\sqrt {5-\sqrt {5}} x}{2\ 2^{3/10}}\right )+c_1 \sin \left (\frac {\sqrt {5+\sqrt {5}} x}{2\ 2^{3/10}}\right )\right ) \]