Internal problem ID [12175]
Internal file name [OUTPUT/10828_Thursday_September_21_2023_05_47_33_AM_87019545/index.tex]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER.
Problems page 172
Problem number: Problem 13.
ODE order: 6.
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]]
\[ \boxed {x^{\left (6\right )}-x^{\prime \prime \prime \prime }=1} \] This is higher order nonhomogeneous ODE. Let the solution be \[ x = x_h + x_p \] Where \(x_h\) is the solution to the homogeneous ODE And \(x_p\) is a particular solution to the nonhomogeneous ODE. \(x_h\) is the solution to \[ x^{\left (6\right )}-x^{\prime \prime \prime \prime } = 0 \] The characteristic equation is \[ \lambda ^{6}-\lambda ^{4} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 0\\ \lambda _3 &= 0\\ \lambda _4 &= 0\\ \lambda _5 &= 1\\ \lambda _6 &= -1 \end {align*}
Therefore the homogeneous solution is \[ x_h(t)={\mathrm e}^{-t} c_{1} +c_{2} +t c_{3} +t^{2} c_{4} +t^{3} c_{5} +{\mathrm e}^{t} c_{6} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} x_1 &= {\mathrm e}^{-t} \\ x_2 &= 1 \\ x_3 &= t \\ x_4 &= t^{2} \\ x_5 &= t^{3} \\ x_6 &= {\mathrm e}^{t} \\ \end{align*} Now the particular solution to the given ODE is found \[ x^{\left (6\right )}-x^{\prime \prime \prime \prime } = 1 \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ 1 \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{1\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{1, t, t^{2}, t^{3}, {\mathrm e}^{t}, {\mathrm e}^{-t}\} \] Since \(1\) is duplicated in the UC_set, then this basis is multiplied by extra \(t\). The UC_set becomes \[ [\{t\}] \] Since \(t\) is duplicated in the UC_set, then this basis is multiplied by extra \(t\). The UC_set becomes \[ [\{t^{2}\}] \] Since \(t^{2}\) is duplicated in the UC_set, then this basis is multiplied by extra \(t\). The UC_set becomes \[ [\{t^{3}\}] \] Since \(t^{3}\) is duplicated in the UC_set, then this basis is multiplied by extra \(t\). The UC_set becomes \[ [\{t^{4}\}] \] Since there was duplication between the basis functions in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis function in the above updated UC_set. \[ x_p = A_{1} t^{4} \] The unknowns \(\{A_{1}\}\) are found by substituting the above trial solution \(x_p\) into the ODE and comparing coefficients. Substituting the trial solution into the ODE and simplifying gives \[ -24 A_{1} = 1 \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = -{\frac {1}{24}}\right ] \] Substituting the above back in the above trial solution \(x_p\), gives the particular solution \[ x_p = -\frac {t^{4}}{24} \] Therefore the general solution is \begin{align*} x &= x_h + x_p \\ &= \left ({\mathrm e}^{-t} c_{1} +c_{2} +t c_{3} +t^{2} c_{4} +t^{3} c_{5} +{\mathrm e}^{t} c_{6}\right ) + \left (-\frac {t^{4}}{24}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} x &= {\mathrm e}^{-t} c_{1} +c_{2} +t c_{3} +t^{2} c_{4} +t^{3} c_{5} +{\mathrm e}^{t} c_{6} -\frac {t^{4}}{24} \\ \end{align*}
Verification of solutions
\[ x = {\mathrm e}^{-t} c_{1} +c_{2} +t c_{3} +t^{2} c_{4} +t^{3} c_{5} +{\mathrm e}^{t} c_{6} -\frac {t^{4}}{24} \] Verified OK.
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 6; linear nonhomogeneous with symmetry [0,1] -> Calling odsolve with the ODE`, diff(diff(_b(_a), _a), _a) = _b(_a)+1, _b(_a)` *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 2; linear nonhomogeneous with symmetry [0,1] trying a double symmetry of the form [xi=0, eta=F(x)] -> Try solving first the homogeneous part of the ODE checking if the LODE has constant coefficients <- constant coefficients successful <- solving first the homogeneous part of the ODE successful <- differential order: 6; linear nonhomogeneous with symmetry [0,1] successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 36
dsolve(diff(x(t),t$6)-diff(x(t),t$4)=1,x(t), singsol=all)
\[ x \left (t \right ) = -\frac {t^{4}}{24}+{\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{t}+\frac {c_{3} t^{3}}{6}+\frac {c_{4} t^{2}}{2}+c_{5} t +c_{6} \]
✓ Solution by Mathematica
Time used: 0.103 (sec). Leaf size: 45
DSolve[x''''''[t]-x''''[t]==1,x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to -\frac {t^4}{24}+c_6 t^3+c_5 t^2+c_4 t+c_1 e^t+c_2 e^{-t}+c_3 \]