Internal problem ID [2759]
Internal file name [OUTPUT/2251_Sunday_June_05_2022_02_56_26_AM_96941610/index.tex]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth
edition, 2015
Section: Chapter 8, Linear differential equations of order n. Section 8.3, The Method of
Undetermined Coefficients. page 525
Problem number: Problem 40.
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, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime \prime }+y^{\prime \prime }+y^{\prime }+y=4 x \,{\mathrm e}^{x}} \] 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 \[ y^{\prime \prime \prime }+y^{\prime \prime }+y^{\prime }+y = 0 \] The characteristic equation is \[ \lambda ^{3}+\lambda ^{2}+\lambda +1 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -1\\ \lambda _2 &= i\\ \lambda _3 &= -i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{-x} c_{1} +{\mathrm e}^{-i x} c_{2} +{\mathrm e}^{i x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{-x} \\ y_2 &= {\mathrm e}^{-i x} \\ y_3 &= {\mathrm e}^{i x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }+y^{\prime \prime }+y^{\prime }+y = 4 x \,{\mathrm e}^{x} \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ 4 x \,{\mathrm e}^{x} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{x \,{\mathrm e}^{x}, {\mathrm e}^{x}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{{\mathrm e}^{i x}, {\mathrm e}^{-x}, {\mathrm e}^{-i x}\} \] 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. \[ y_p = A_{1} x \,{\mathrm e}^{x}+A_{2} {\mathrm e}^{x} \] The unknowns \(\{A_{1}, A_{2}\}\) 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 \[ 6 A_{1} {\mathrm e}^{x}+4 A_{1} x \,{\mathrm e}^{x}+4 A_{2} {\mathrm e}^{x} = 4 x \,{\mathrm e}^{x} \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = 1, A_{2} = -{\frac {3}{2}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = x \,{\mathrm e}^{x}-\frac {3 \,{\mathrm e}^{x}}{2} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{-x} c_{1} +{\mathrm e}^{-i x} c_{2} +{\mathrm e}^{i x} c_{3}\right ) + \left (x \,{\mathrm e}^{x}-\frac {3 \,{\mathrm e}^{x}}{2}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-x} c_{1} +{\mathrm e}^{-i x} c_{2} +{\mathrm e}^{i x} c_{3} +x \,{\mathrm e}^{x}-\frac {3 \,{\mathrm e}^{x}}{2} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{-x} c_{1} +{\mathrm e}^{-i x} c_{2} +{\mathrm e}^{i x} c_{3} +x \,{\mathrm e}^{x}-\frac {3 \,{\mathrm e}^{x}}{2} \] 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: 0.016 (sec). Leaf size: 27
dsolve(diff(y(x),x$3)+diff(y(x),x$2)+diff(y(x),x)+y(x)=4*x*exp(x),y(x), singsol=all)
\[ y \left (x \right ) = c_{3} {\mathrm e}^{-x}+\cos \left (x \right ) c_{1} +x \,{\mathrm e}^{x}+\sin \left (x \right ) c_{2} -\frac {3 \,{\mathrm e}^{x}}{2} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 36
DSolve[y'''[x]+y''[x]+y'[x]+y[x]==4*x*Exp[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^x x-\frac {3 e^x}{2}+c_3 e^{-x}+c_1 \cos (x)+c_2 \sin (x) \]