Internal problem ID [12768]
Internal file name [OUTPUT/11421_Friday_November_03_2023_06_32_47_AM_79775332/index.tex]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 4. N-th Order Linear Differential Equations. Exercises 4.3, page 210
Problem number: 10.
ODE order: 8.
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]]
\[ \boxed {y^{\left (8\right )}+8 y^{\prime \prime \prime \prime }+16 y=0} \] The characteristic equation is \[ \lambda ^{8}+8 \lambda ^{4}+16 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 1-i\\ \lambda _2 &= 1+i\\ \lambda _3 &= -1-i\\ \lambda _4 &= -1+i\\ \lambda _5 &= 1-i\\ \lambda _6 &= 1+i\\ \lambda _7 &= -1-i\\ \lambda _8 &= -1+i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (1+i\right ) x} c_{1} +x \,{\mathrm e}^{\left (1+i\right ) x} c_{2} +{\mathrm e}^{\left (-1+i\right ) x} c_{3} +x \,{\mathrm e}^{\left (-1+i\right ) x} c_{4} +{\mathrm e}^{\left (-1-i\right ) x} c_{5} +x \,{\mathrm e}^{\left (-1-i\right ) x} c_{6} +{\mathrm e}^{\left (1-i\right ) x} c_{7} +x \,{\mathrm e}^{\left (1-i\right ) x} c_{8} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{\left (1+i\right ) x}\\ y_2 &= x \,{\mathrm e}^{\left (1+i\right ) x}\\ y_3 &= {\mathrm e}^{\left (-1+i\right ) x}\\ y_4 &= x \,{\mathrm e}^{\left (-1+i\right ) x}\\ y_5 &= {\mathrm e}^{\left (-1-i\right ) x}\\ y_6 &= x \,{\mathrm e}^{\left (-1-i\right ) x}\\ y_7 &= {\mathrm e}^{\left (1-i\right ) x}\\ y_8 &= x \,{\mathrm e}^{\left (1-i\right ) x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\left (1+i\right ) x} c_{1} +x \,{\mathrm e}^{\left (1+i\right ) x} c_{2} +{\mathrm e}^{\left (-1+i\right ) x} c_{3} +x \,{\mathrm e}^{\left (-1+i\right ) x} c_{4} +{\mathrm e}^{\left (-1-i\right ) x} c_{5} +x \,{\mathrm e}^{\left (-1-i\right ) x} c_{6} +{\mathrm e}^{\left (1-i\right ) x} c_{7} +x \,{\mathrm e}^{\left (1-i\right ) x} c_{8} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{\left (1+i\right ) x} c_{1} +x \,{\mathrm e}^{\left (1+i\right ) x} c_{2} +{\mathrm e}^{\left (-1+i\right ) x} c_{3} +x \,{\mathrm e}^{\left (-1+i\right ) x} c_{4} +{\mathrm e}^{\left (-1-i\right ) x} c_{5} +x \,{\mathrm e}^{\left (-1-i\right ) x} c_{6} +{\mathrm e}^{\left (1-i\right ) x} c_{7} +x \,{\mathrm e}^{\left (1-i\right ) x} c_{8} \] 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: 47
dsolve(diff(y(x),x$8)+8*diff(y(x),x$4)+16*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (\left (c_{4} x +c_{2} \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_{3} x +c_{1} \right )\right ) {\mathrm e}^{-x}+{\mathrm e}^{x} \left (\left (c_{8} x +c_{6} \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_{7} x +c_{5} \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 66
DSolve[D[y[x],{x,8}]+8*y''''[x]+16*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-x} \left (\left (c_4 x+c_7 e^{2 x}+c_8 e^{2 x} x+c_3\right ) \cos (x)+\left (c_2 x+c_5 e^{2 x}+c_6 e^{2 x} x+c_1\right ) \sin (x)\right ) \]