Internal problem ID [4600]
Internal file name [OUTPUT/4093_Sunday_June_05_2022_12_21_21_PM_91188637/index.tex]
Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 4. Higher order linear differential equations. Lesson 20. Constant
coefficients
Problem number: Exercise 20.30, page 220.
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]]
\[ \boxed {y^{\left (5\right )}+2 y^{\prime \prime \prime }+y^{\prime }=0} \] The characteristic equation is \[ \lambda ^{5}+2 \lambda ^{3}+\lambda = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= i\\ \lambda _3 &= -i\\ \lambda _4 &= i\\ \lambda _5 &= -i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} +{\mathrm e}^{i x} c_{2} +x \,{\mathrm e}^{i x} c_{3} +{\mathrm e}^{-i x} c_{4} +x \,{\mathrm e}^{-i x} c_{5} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= 1\\ y_2 &= {\mathrm e}^{i x}\\ y_3 &= {\mathrm e}^{i x} x\\ y_4 &= {\mathrm e}^{-i x}\\ y_5 &= {\mathrm e}^{-i x} x \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} +{\mathrm e}^{i x} c_{2} +x \,{\mathrm e}^{i x} c_{3} +{\mathrm e}^{-i x} c_{4} +x \,{\mathrm e}^{-i x} c_{5} \\ \end{align*}
Verification of solutions
\[ y = c_{1} +{\mathrm e}^{i x} c_{2} +x \,{\mathrm e}^{i x} c_{3} +{\mathrm e}^{-i x} c_{4} +x \,{\mathrm e}^{-i x} c_{5} \] 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: 22
dsolve(diff(y(x),x$5)+2*diff(y(x),x$3)+diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{5} x +c_{3} \right ) \cos \left (x \right )+\left (x c_{4} +c_{2} \right ) \sin \left (x \right )+c_{1} \]
✓ Solution by Mathematica
Time used: 0.05 (sec). Leaf size: 35
DSolve[y'''''[x]+2*y'''[x]+y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (-c_4 x+c_2-c_3) \cos (x)+(c_2 x+c_1+c_4) \sin (x)+c_5 \]