Internal problem ID [12176]
Internal file name [OUTPUT/10829_Thursday_September_21_2023_05_47_34_AM_26296989/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 14.
ODE order: 4.
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, _with_linear_symmetries]]
\[ \boxed {x^{\prime \prime \prime \prime }-2 x^{\prime \prime }+x=t^{2}-3} \] 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 \[ x^{\prime \prime \prime \prime }-2 x^{\prime \prime }+x = 0 \] The characteristic equation is \[ \lambda ^{4}-2 \lambda ^{2}+1 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 1\\ \lambda _2 &= 1\\ \lambda _3 &= -1\\ \lambda _4 &= -1 \end {align*}
Therefore the homogeneous solution is \[ x_h(t)={\mathrm e}^{-t} c_{1} +t \,{\mathrm e}^{-t} c_{2} +{\mathrm e}^{t} c_{3} +t \,{\mathrm e}^{t} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} x_1 &= {\mathrm e}^{-t} \\ x_2 &= t \,{\mathrm e}^{-t} \\ x_3 &= {\mathrm e}^{t} \\ x_4 &= t \,{\mathrm e}^{t} \\ \end{align*} Now the particular solution to the given ODE is found \[ x^{\prime \prime \prime \prime }-2 x^{\prime \prime }+x = t^{2}-3 \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ t^{2}+1 \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{1, t, t^{2}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{t \,{\mathrm e}^{t}, t \,{\mathrm e}^{-t}, {\mathrm e}^{t}, {\mathrm e}^{-t}\} \] Since there is no duplication between the basis function in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis in the UC_set. \[ x_p = A_{3} t^{2}+A_{2} t +A_{1} \] The unknowns \(\{A_{1}, A_{2}, A_{3}\}\) 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 \[ A_{3} t^{2}+A_{2} t +A_{1}-4 A_{3} = t^{2}-3 \] Solving for the unknowns by comparing coefficients results in \[ [A_{1} = 1, A_{2} = 0, A_{3} = 1] \] Substituting the above back in the above trial solution \(x_p\), gives the particular solution \[ x_p = t^{2}+1 \] Therefore the general solution is \begin{align*} x &= x_h + x_p \\ &= \left ({\mathrm e}^{-t} c_{1} +t \,{\mathrm e}^{-t} c_{2} +{\mathrm e}^{t} c_{3} +t \,{\mathrm e}^{t} c_{4}\right ) + \left (t^{2}+1\right ) \\ \end{align*} Which simplifies to \[ x = \left (c_{2} t +c_{1} \right ) {\mathrm e}^{-t}+{\mathrm e}^{t} \left (c_{4} t +c_{3} \right )+t^{2}+1 \]
The solution(s) found are the following \begin{align*} \tag{1} x &= \left (c_{2} t +c_{1} \right ) {\mathrm e}^{-t}+{\mathrm e}^{t} \left (c_{4} t +c_{3} \right )+t^{2}+1 \\ \end{align*}
Verification of solutions
\[ x = \left (c_{2} t +c_{1} \right ) {\mathrm e}^{-t}+{\mathrm e}^{t} \left (c_{4} t +c_{3} \right )+t^{2}+1 \] 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: 4; linear nonhomogeneous with symmetry [0,1] trying high order linear exact nonhomogeneous trying differential order: 4; missing the dependent variable checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 27
dsolve(diff(x(t),t$4)-2*diff(x(t),t$2)+x(t)=t^2-3,x(t), singsol=all)
\[ x \left (t \right ) = \left (c_{4} t +c_{2} \right ) {\mathrm e}^{-t}+\left (c_{3} t +c_{1} \right ) {\mathrm e}^{t}+t^{2}+1 \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 38
DSolve[x''''[t]-2*x''[t]+x[t]==t^2-3,x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to t^2+c_2 e^{-t} t+c_1 e^{-t}+e^t (c_4 t+c_3)+1 \]