Internal problem ID [5861]
Internal file name [OUTPUT/5109_Sunday_June_05_2022_03_24_54_PM_13785063/index.tex]
Book: THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS.
K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section: Chapter 3. Ordinary Differential Equations. Section 3.5 HIGHER ORDER ODE. Page
181
Problem number: Example 3.34.
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, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y=4 \,{\mathrm e}^{t}} \] 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 \[ y_h(t)={\mathrm e}^{t} c_{1} +t \,{\mathrm e}^{t} c_{2} +t^{2} {\mathrm e}^{t} c_{3} \] 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 \[ y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y = 4 \,{\mathrm e}^{t} \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ 4 \,{\mathrm e}^{t} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{{\mathrm e}^{t}\}] \] 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 \({\mathrm e}^{t}\) is duplicated in the UC_set, then this basis is multiplied by extra \(t\). The UC_set becomes \[ [\{t \,{\mathrm e}^{t}\}] \] Since \(t \,{\mathrm e}^{t}\) is duplicated in the UC_set, then this basis is multiplied by extra \(t\). The UC_set becomes \[ [\{t^{2} {\mathrm e}^{t}\}] \] Since \(t^{2} {\mathrm e}^{t}\) is duplicated in the UC_set, then this basis is multiplied by extra \(t\). The UC_set becomes \[ [\{t^{3} {\mathrm e}^{t}\}] \] 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. \[ y_p = A_{1} t^{3} {\mathrm e}^{t} \] 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 \[ 6 A_{1} {\mathrm e}^{t} = 4 \,{\mathrm e}^{t} \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = {\frac {2}{3}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {2 t^{3} {\mathrm e}^{t}}{3} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{t} c_{1} +t \,{\mathrm e}^{t} c_{2} +t^{2} {\mathrm e}^{t} c_{3}\right ) + \left (\frac {2 t^{3} {\mathrm e}^{t}}{3}\right ) \\ \end{align*} Which simplifies to \[ y = {\mathrm e}^{t} \left (c_{3} t^{2}+c_{2} t +c_{1} \right )+\frac {2 t^{3} {\mathrm e}^{t}}{3} \]
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 )+\frac {2 t^{3} {\mathrm e}^{t}}{3} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{t} \left (c_{3} t^{2}+c_{2} t +c_{1} \right )+\frac {2 t^{3} {\mathrm e}^{t}}{3} \] 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: 0.015 (sec). Leaf size: 22
dsolve(diff(y(t),t$3)-3*diff(y(t),t$2)+3*diff(y(t),t)-y(t)=4*exp(t),y(t), singsol=all)
\[ y \left (t \right ) = {\mathrm e}^{t} \left (\frac {2}{3} t^{3}+c_{1} +t c_{2} +t^{2} c_{3} \right ) \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 34
DSolve[y'''[t]-3*y''[t]+3*y'[t]-y[t]==4*Exp[t],y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {1}{3} e^t \left (2 t^3+3 c_3 t^2+3 c_2 t+3 c_1\right ) \]