Internal problem ID [11504]
Internal file name [OUTPUT/10487_Thursday_May_18_2023_04_20_52_AM_35911259/index.tex]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag,
NY. 2015.
Section: Chapter 2, Second order linear equations. Section 2.5 Higher order equations. Exercises
page 130
Problem number: 1(e).
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_y]]
\[ \boxed {x^{\prime \prime \prime }+x^{\prime \prime }=2 \,{\mathrm e}^{t}+3 t^{2}} \] 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 \[ x^{\prime \prime \prime }+x^{\prime \prime } = 0 \] The characteristic equation is \[ \lambda ^{3}+\lambda ^{2} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -1\\ \lambda _2 &= 0\\ \lambda _3 &= 0 \end {align*}
Therefore the homogeneous solution is \[ x_h(t)=c_{1} {\mathrm e}^{-t}+c_{2} +c_{3} t \] 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 \\ \end{align*} Now the particular solution to the given ODE is found \[ x^{\prime \prime \prime }+x^{\prime \prime } = 2 \,{\mathrm e}^{t}+3 t^{2} \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ 2 \,{\mathrm e}^{t}+3 t^{2} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{{\mathrm e}^{t}\}, \{1, t, t^{2}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{1, t, {\mathrm e}^{-t}\} \] Since \(1\) is duplicated in the UC_set, then this basis is multiplied by extra \(t\). The UC_set becomes \[ [\{{\mathrm e}^{t}\}, \{t, t^{2}, t^{3}\}] \] Since \(t\) is duplicated in the UC_set, then this basis is multiplied by extra \(t\). The UC_set becomes \[ [\{{\mathrm e}^{t}\}, \{t^{2}, t^{3}, t^{4}\}] \] 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. \[ x_p = A_{1} {\mathrm e}^{t}+A_{2} t^{2}+A_{3} t^{3}+A_{4} t^{4} \] The unknowns \(\{A_{1}, A_{2}, A_{3}, A_{4}\}\) 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 \[ 2 A_{1} {\mathrm e}^{t}+6 A_{3}+24 A_{4} t +2 A_{2}+6 A_{3} t +12 A_{4} t^{2} = 2 \,{\mathrm e}^{t}+3 t^{2} \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = 1, A_{2} = 3, A_{3} = -1, A_{4} = {\frac {1}{4}}\right ] \] Substituting the above back in the above trial solution \(x_p\), gives the particular solution \[ x_p = {\mathrm e}^{t}+3 t^{2}-t^{3}+\frac {t^{4}}{4} \] Therefore the general solution is \begin{align*} x &= x_h + x_p \\ &= \left (c_{1} {\mathrm e}^{-t}+c_{2} +c_{3} t\right ) + \left ({\mathrm e}^{t}+3 t^{2}-t^{3}+\frac {t^{4}}{4}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} x &= c_{1} {\mathrm e}^{-t}+c_{2} +c_{3} t +{\mathrm e}^{t}+3 t^{2}-t^{3}+\frac {t^{4}}{4} \\ \end{align*}
Verification of solutions
\[ x = c_{1} {\mathrm e}^{-t}+c_{2} +c_{3} t +{\mathrm e}^{t}+3 t^{2}-t^{3}+\frac {t^{4}}{4} \] Verified OK.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable -> Calling odsolve with the ODE`, diff(_b(_a), _a) = 3*_a^2+2*exp(_a)-_b(_a), _b(_a)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful <- high order exact linear fully integrable successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 32
dsolve(diff(x(t),t$3)+diff(x(t),t$2)=2*exp(t)+3*t^2,x(t), singsol=all)
\[ x \left (t \right ) = \frac {t^{4}}{4}+3 t^{2}-t^{3}+{\mathrm e}^{-t} c_{1} +{\mathrm e}^{t}+c_{2} t +c_{3} \]
✓ Solution by Mathematica
Time used: 0.308 (sec). Leaf size: 40
DSolve[x'''[t]+x''[t]==2*Exp[t]+3*t^2,x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \frac {t^4}{4}-t^3+3 t^2+e^t+c_3 t+c_1 e^{-t}+c_2 \]