Internal problem ID [2795]
Internal file name [OUTPUT/2287_Sunday_June_05_2022_02_57_49_AM_20383296/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 8.7, The Variation of
Parameters Method. page 556
Problem number: Problem 22.
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]]
\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+9 y^{\prime }=12 \,{\mathrm e}^{3 x}} \] 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 \[ y^{\prime \prime \prime }-6 y^{\prime \prime }+9 y^{\prime } = 0 \] The characteristic equation is \[ \lambda ^{3}-6 \lambda ^{2}+9 \lambda = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 3\\ \lambda _3 &= 3 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} +c_{2} {\mathrm e}^{3 x}+x \,{\mathrm e}^{3 x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= {\mathrm e}^{3 x} \\ y_3 &= x \,{\mathrm e}^{3 x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }-6 y^{\prime \prime }+9 y^{\prime } = 12 \,{\mathrm e}^{3 x} \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ 12 \,{\mathrm e}^{3 x} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{{\mathrm e}^{3 x}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{1, x \,{\mathrm e}^{3 x}, {\mathrm e}^{3 x}\} \] Since \({\mathrm e}^{3 x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x \,{\mathrm e}^{3 x}\}] \] Since \(x \,{\mathrm e}^{3 x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{{\mathrm e}^{3 x} x^{2}\}] \] 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. \[ y_p = A_{1} {\mathrm e}^{3 x} x^{2} \] 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 \[ 6 A_{1} {\mathrm e}^{3 x} = 12 \,{\mathrm e}^{3 x} \] Solving for the unknowns by comparing coefficients results in \[ [A_{1} = 2] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = 2 \,{\mathrm e}^{3 x} x^{2} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} +c_{2} {\mathrm e}^{3 x}+x \,{\mathrm e}^{3 x} c_{3}\right ) + \left (2 \,{\mathrm e}^{3 x} x^{2}\right ) \\ \end{align*} Which simplifies to \[ y = \left (c_{3} x +c_{2} \right ) {\mathrm e}^{3 x}+c_{1} +2 \,{\mathrm e}^{3 x} x^{2} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \left (c_{3} x +c_{2} \right ) {\mathrm e}^{3 x}+c_{1} +2 \,{\mathrm e}^{3 x} x^{2} \\ \end{align*}
Verification of solutions
\[ y = \left (c_{3} x +c_{2} \right ) {\mathrm e}^{3 x}+c_{1} +2 \,{\mathrm e}^{3 x} x^{2} \] 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] -> Calling odsolve with the ODE`, diff(diff(_b(_a), _a), _a) = 6*(diff(_b(_a), _a))-9*_b(_a)+12*exp(3*_a), _b(_a)` *** Sublevel 2 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: 31
dsolve(diff(y(x),x$3)-6*diff(y(x),x$2)+9*diff(y(x),x)=12*exp(3*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (4+18 x^{2}+3 \left (-4+c_{1} \right ) x -c_{1} +3 c_{2} \right ) {\mathrm e}^{3 x}}{9}+c_{3} \]
✓ Solution by Mathematica
Time used: 0.06 (sec). Leaf size: 39
DSolve[y'''[x]-6*y''[x]+9*y'[x]==12*Exp[3*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{9} e^{3 x} \left (18 x^2+3 (-4+c_2) x+4+3 c_1-c_2\right )+c_3 \]