Internal problem ID [12358]
Internal file name [OUTPUT/11011_Monday_October_02_2023_02_48_01_AM_59068219/index.tex]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A.
Dobrushkin. CRC Press 2015
Section: Chapter 6. Introduction to Systems of ODEs. Problems page 408
Problem number: Problem 2(d).
ODE order: 3.
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
[[_3rd_order, _missing_x]]
\[ \boxed {y^{\prime \prime \prime }+3 y^{\prime \prime }+3 y^{\prime }+y=5} \] This is higher order nonhomogeneous ODE. Let the solution be \[ y = y_h + y_p \] Where \(y_h\) is the solution to the homogeneous ODE And \(y_p\) is a particular solution to the nonhomogeneous ODE. \(y_h\) is the solution to \[ y^{\prime \prime \prime }+3 y^{\prime \prime }+3 y^{\prime }+y = 0 \] The characteristic equation is \[ \lambda ^{3}+3 \lambda ^{2}+3 \lambda +1 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -1\\ \lambda _2 &= -1\\ \lambda _3 &= -1 \end {align*}
Therefore the homogeneous solution is \[ y_h(t)=c_{1} {\mathrm e}^{-t}+c_{2} t \,{\mathrm e}^{-t}+t^{2} {\mathrm e}^{-t} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{-t} \\ y_2 &= t \,{\mathrm e}^{-t} \\ y_3 &= t^{2} {\mathrm e}^{-t} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }+3 y^{\prime \prime }+3 y^{\prime }+y = 5 \] 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 \[ \{t \,{\mathrm e}^{-t}, t^{2} {\mathrm e}^{-t}, {\mathrm e}^{-t}\} \] 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. \[ y_p = A_{1} \] The unknowns \(\{A_{1}\}\) are found by substituting the above trial solution \(y_p\) into the ODE and comparing coefficients. Substituting the trial solution into the ODE and simplifying gives \[ A_{1} = 5 \] Solving for the unknowns by comparing coefficients results in \[ [A_{1} = 5] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = 5 \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} {\mathrm e}^{-t}+c_{2} t \,{\mathrm e}^{-t}+t^{2} {\mathrm e}^{-t} c_{3}\right ) + \left (5\right ) \\ \end{align*} Which simplifies to \[ y = {\mathrm e}^{-t} \left (c_{3} t^{2}+c_{2} t +c_{1} \right )+5 \]
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-t} \left (c_{3} t^{2}+c_{2} t +c_{1} \right )+5 \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{-t} \left (c_{3} t^{2}+c_{2} t +c_{1} \right )+5 \] Verified OK.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 3; linear nonhomogeneous with symmetry [0,1] trying high order linear exact nonhomogeneous trying differential order: 3; missing the dependent variable checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 21
dsolve(diff(y(t),t$3)+3*diff(y(t),t$2)+3*diff(y(t),t)+y(t)=5,y(t), singsol=all)
\[ y \left (t \right ) = 5+\left (c_{3} t^{2}+c_{2} t +c_{1} \right ) {\mathrm e}^{-t} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 28
DSolve[y'''[t]+3*y''[t]+3*y'[t]+y[t]==5,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to e^{-t} \left (5 e^t+t (c_3 t+c_2)+c_1\right ) \]