Internal problem ID [1560]
Internal file name [OUTPUT/1561_Sunday_June_05_2022_02_22_28_AM_73161/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 9 Introduction to Linear Higher Order Equations. Section 9.3. Undetermined
Coefficients for Higher Order Equations. Page 495
Problem number: section 9.3, problem 63.
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]]
\[ \boxed {y^{\prime \prime \prime }+2 y^{\prime \prime }+y^{\prime }=-2 \,{\mathrm e}^{-x} \left (6 x^{2}-18 x +7\right )} \] 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 \[ y^{\prime \prime \prime }+2 y^{\prime \prime }+y^{\prime } = 0 \] The characteristic equation is \[ \lambda ^{3}+2 \lambda ^{2}+\lambda = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= -1\\ \lambda _3 &= -1 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{-x}+c_{2} x \,{\mathrm e}^{-x}+c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{-x} \\ y_2 &= x \,{\mathrm e}^{-x} \\ y_3 &= 1 \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }+2 y^{\prime \prime }+y^{\prime } = -2 \,{\mathrm e}^{-x} \left (6 x^{2}-18 x +7\right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ -2 \,{\mathrm e}^{-x} \left (6 x^{2}-18 x +7\right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{x \,{\mathrm e}^{-x}, x^{2} {\mathrm e}^{-x}, {\mathrm e}^{-x}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{1, x \,{\mathrm e}^{-x}, {\mathrm e}^{-x}\} \] Since \({\mathrm e}^{-x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x \,{\mathrm e}^{-x}, x^{2} {\mathrm e}^{-x}, {\mathrm e}^{-x} x^{3}\}] \] Since \(x \,{\mathrm e}^{-x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x^{2} {\mathrm e}^{-x}, {\mathrm e}^{-x} x^{3}, {\mathrm e}^{-x} x^{4}\}] \] 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. \[ y_p = A_{1} x^{2} {\mathrm e}^{-x}+A_{2} {\mathrm e}^{-x} x^{3}+A_{3} {\mathrm e}^{-x} x^{4} \] The unknowns \(\{A_{1}, A_{2}, A_{3}\}\) 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 \[ 6 A_{2} {\mathrm e}^{-x}-6 A_{2} {\mathrm e}^{-x} x -12 A_{3} {\mathrm e}^{-x} x^{2}+24 A_{3} {\mathrm e}^{-x} x -2 A_{1} {\mathrm e}^{-x} = -2 \,{\mathrm e}^{-x} \left (6 x^{2}-18 x +7\right ) \] Solving for the unknowns by comparing coefficients results in \[ [A_{1} = 1, A_{2} = -2, A_{3} = 1] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = x^{2} {\mathrm e}^{-x}-2 \,{\mathrm e}^{-x} x^{3}+{\mathrm e}^{-x} x^{4} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} {\mathrm e}^{-x}+c_{2} x \,{\mathrm e}^{-x}+c_{3}\right ) + \left (x^{2} {\mathrm e}^{-x}-2 \,{\mathrm e}^{-x} x^{3}+{\mathrm e}^{-x} x^{4}\right ) \\ \end{align*} Which simplifies to \[ y = \left (c_{2} x +c_{1} \right ) {\mathrm e}^{-x}+c_{3} +x^{2} {\mathrm e}^{-x}-2 \,{\mathrm e}^{-x} x^{3}+{\mathrm e}^{-x} x^{4} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \left (c_{2} x +c_{1} \right ) {\mathrm e}^{-x}+c_{3} +x^{2} {\mathrm e}^{-x}-2 \,{\mathrm e}^{-x} x^{3}+{\mathrm e}^{-x} x^{4} \\ \end{align*}
Verification of solutions
\[ y = \left (c_{2} x +c_{1} \right ) {\mathrm e}^{-x}+c_{3} +x^{2} {\mathrm e}^{-x}-2 \,{\mathrm e}^{-x} x^{3}+{\mathrm e}^{-x} x^{4} \] 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] -> Calling odsolve with the ODE`, diff(diff(_b(_a), _a), _a) = -12*exp(-_a)*_a^2+36*_a*exp(-_a)-14*exp(-_a)-_b(_a)-2*(diff(_b(_a), _ Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 2; linear nonhomogeneous with symmetry [0,1] trying a double symmetry of the form [xi=0, eta=F(x)] -> Try solving first the homogeneous part of the ODE checking if the LODE has constant coefficients <- constant coefficients successful <- solving first the homogeneous part of the ODE successful <- differential order: 3; linear nonhomogeneous with symmetry [0,1] successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 37
dsolve(diff(y(x),x$3)+2*diff(y(x),x$2)+1*diff(y(x),x)-0*y(x)=-2*exp(-x)*(7-18*x+6*x^2),y(x), singsol=all)
\[ y \left (x \right ) = \left (x^{4}-2 x^{3}+x^{2}+\left (-c_{1} +2\right ) x -c_{1} -c_{2} +2\right ) {\mathrm e}^{-x}+c_{3} \]
✓ Solution by Mathematica
Time used: 0.101 (sec). Leaf size: 42
DSolve[y'''[x]+2*y''[x]+1*y'[x]-0*y[x]==-2*Exp[-x]*(7-18*x+6*x^2),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-x} \left (x^4-2 x^3+x^2-(-2+c_2) x+2-c_1-c_2\right )+c_3 \]