Internal problem ID [13621]
Internal file name [OUTPUT/12793_Saturday_February_17_2024_08_44_51_AM_10851037/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 19. Arbitrary Homogeneous linear equations with constant coefficients.
Additional exercises page 369
Problem number: 19.1 (f).
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]]
\[ \boxed {y^{\left (5\right )}+18 y^{\prime \prime \prime }+81 y^{\prime }=0} \] The characteristic equation is \[ \lambda ^{5}+18 \lambda ^{3}+81 \lambda = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 3 i\\ \lambda _3 &= -3 i\\ \lambda _4 &= 3 i\\ \lambda _5 &= -3 i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} +{\mathrm e}^{-3 i x} c_{2} +x \,{\mathrm e}^{-3 i x} c_{3} +{\mathrm e}^{3 i x} c_{4} +x \,{\mathrm e}^{3 i x} c_{5} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= 1\\ y_2 &= {\mathrm e}^{-3 i x}\\ y_3 &= x \,{\mathrm e}^{-3 i x}\\ y_4 &= {\mathrm e}^{3 i x}\\ y_5 &= x \,{\mathrm e}^{3 i x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} +{\mathrm e}^{-3 i x} c_{2} +x \,{\mathrm e}^{-3 i x} c_{3} +{\mathrm e}^{3 i x} c_{4} +x \,{\mathrm e}^{3 i x} c_{5} \\ \end{align*}
Verification of solutions
\[ y = c_{1} +{\mathrm e}^{-3 i x} c_{2} +x \,{\mathrm e}^{-3 i x} c_{3} +{\mathrm e}^{3 i x} c_{4} +x \,{\mathrm e}^{3 i x} c_{5} \] 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: 26
dsolve(diff(y(x),x$5)+18*diff(y(x),x$3)+81*diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{5} x +c_{3} \right ) \cos \left (3 x \right )+\left (c_{4} x +c_{2} \right ) \sin \left (3 x \right )+c_{1} \]
✓ Solution by Mathematica
Time used: 0.11 (sec). Leaf size: 48
DSolve[y'''''[x]+18*y'''[x]+81*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{9} ((c_2-3 (c_4 x+c_3)) \cos (3 x)+(3 c_2 x+3 c_1+c_4) \sin (3 x)+9 c_5) \]