Internal problem ID [1507]
Internal file name [OUTPUT/1508_Sunday_June_05_2022_02_20_02_AM_9230329/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 10.
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]]
\[ \boxed {y^{\prime \prime \prime }-5 y^{\prime \prime }+3 y^{\prime }+9 y=2 \,{\mathrm e}^{3 x} \left (11-24 x \right )} \] 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 \[ y^{\prime \prime \prime }-5 y^{\prime \prime }+3 y^{\prime }+9 y = 0 \] The characteristic equation is \[ \lambda ^{3}-5 \lambda ^{2}+3 \lambda +9 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -1\\ \lambda _2 &= 3\\ \lambda _3 &= 3 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{-x} c_{1} +{\mathrm e}^{3 x} c_{2} +x \,{\mathrm e}^{3 x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{-x} \\ y_2 &= {\mathrm e}^{3 x} \\ y_3 &= x \,{\mathrm e}^{3 x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }-5 y^{\prime \prime }+3 y^{\prime }+9 y = 2 \,{\mathrm e}^{3 x} \left (11-24 x \right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ 2 \,{\mathrm e}^{3 x} \left (11-24 x \right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{x \,{\mathrm e}^{3 x}, {\mathrm e}^{3 x}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{x \,{\mathrm e}^{3 x}, {\mathrm e}^{-x}, {\mathrm e}^{3 x}\} \] Since \({\mathrm e}^{3 x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x \,{\mathrm e}^{3 x}, x^{2} {\mathrm e}^{3 x}\}] \] Since \(x \,{\mathrm e}^{3 x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x^{2} {\mathrm e}^{3 x}, x^{3} {\mathrm e}^{3 x}\}] \] 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. \[ y_p = A_{1} x^{2} {\mathrm e}^{3 x}+A_{2} x^{3} {\mathrm e}^{3 x} \] 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 \[ 8 A_{1} {\mathrm e}^{3 x}+6 A_{2} {\mathrm e}^{3 x}+24 A_{2} x \,{\mathrm e}^{3 x} = 2 \,{\mathrm e}^{3 x} \left (11-24 x \right ) \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = {\frac {17}{4}}, A_{2} = -2\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {17 x^{2} {\mathrm e}^{3 x}}{4}-2 x^{3} {\mathrm e}^{3 x} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{-x} c_{1} +{\mathrm e}^{3 x} c_{2} +x \,{\mathrm e}^{3 x} c_{3}\right ) + \left (\frac {17 x^{2} {\mathrm e}^{3 x}}{4}-2 x^{3} {\mathrm e}^{3 x}\right ) \\ \end{align*} Which simplifies to \[ y = \left (c_{3} x +c_{2} \right ) {\mathrm e}^{3 x}+{\mathrm e}^{-x} c_{1} +\frac {17 x^{2} {\mathrm e}^{3 x}}{4}-2 x^{3} {\mathrm e}^{3 x} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \left (c_{3} x +c_{2} \right ) {\mathrm e}^{3 x}+{\mathrm e}^{-x} c_{1} +\frac {17 x^{2} {\mathrm e}^{3 x}}{4}-2 x^{3} {\mathrm e}^{3 x} \\ \end{align*}
Verification of solutions
\[ y = \left (c_{3} x +c_{2} \right ) {\mathrm e}^{3 x}+{\mathrm e}^{-x} c_{1} +\frac {17 x^{2} {\mathrm e}^{3 x}}{4}-2 x^{3} {\mathrm e}^{3 x} \] 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: 35
dsolve(diff(y(x),x$3)-5*diff(y(x),x$2)+3*diff(y(x),x)+9*y(x)=2*exp(3*x)*(11-24*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-8 x^{3}+4 c_{3} x +17 x^{2}+4 c_{2} \right ) {\mathrm e}^{3 x}}{4}+{\mathrm e}^{-x} c_{1} \]
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 46
DSolve[y'''[x]-5*y''[x]+3*y'[x]+9*y[x]==2*Exp[3*x]*(11-24*x),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{3 x} \left (-2 x^3+\frac {17 x^2}{4}+\left (-\frac {17}{8}+c_3\right ) x+\frac {17}{32}+c_2\right )+c_1 e^{-x} \]