11.2.8 problem 15

Internal problem ID [1479]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 4.2, Higher order linear differential equations. Constant coefficients. page 180
Problem number : 15
Date solved : Monday, January 27, 2025 at 04:57:48 AM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\left (8\right )}+8 y^{\prime \prime \prime \prime }+16 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 47

dsolve(diff(y(x),x$8)+8*diff(y(x),x$4)+16*y(x)=0,y(x), singsol=all)
 

\[ y = \left (\left (c_4 x +c_2 \right ) \cos \left (x \right )+\sin \left (x \right ) \left (x c_3 +c_1 \right )\right ) {\mathrm e}^{-x}+\left (\left (c_8 x +c_6 \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_7 x +c_5 \right )\right ) {\mathrm e}^{x} \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 238

DSolve[D[y[x],{x,8}]+8*D[y[x],{x,4}]+3*D[y[x],{x,3}]+16*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^8+8 \text {$\#$1}^4+3 \text {$\#$1}^3+16\&,1\right ]\right )+c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^8+8 \text {$\#$1}^4+3 \text {$\#$1}^3+16\&,2\right ]\right )+c_5 \exp \left (x \text {Root}\left [\text {$\#$1}^8+8 \text {$\#$1}^4+3 \text {$\#$1}^3+16\&,5\right ]\right )+c_6 \exp \left (x \text {Root}\left [\text {$\#$1}^8+8 \text {$\#$1}^4+3 \text {$\#$1}^3+16\&,6\right ]\right )+c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^8+8 \text {$\#$1}^4+3 \text {$\#$1}^3+16\&,3\right ]\right )+c_4 \exp \left (x \text {Root}\left [\text {$\#$1}^8+8 \text {$\#$1}^4+3 \text {$\#$1}^3+16\&,4\right ]\right )+c_7 \exp \left (x \text {Root}\left [\text {$\#$1}^8+8 \text {$\#$1}^4+3 \text {$\#$1}^3+16\&,7\right ]\right )+c_8 \exp \left (x \text {Root}\left [\text {$\#$1}^8+8 \text {$\#$1}^4+3 \text {$\#$1}^3+16\&,8\right ]\right ) \]