problem Problem 41

Internal problem ID [2760]
Internal ﬁle name [OUTPUT/2252_Sunday_June_05_2022_02_56_28_AM_61127683/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 41.
ODE order: 4.
ODE degree: 1.

The type(s) of ODE detected by this program : "higher_order_linear_constant_coeﬃcients_ODE"

\[ \boxed {y^{\prime \prime \prime \prime }+104 y^{\prime \prime \prime }+2740 y^{\prime \prime }=5 \,{\mathrm e}^{-2 x} \cos \left (3 x \right )} \] 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 \[ y^{\prime \prime \prime \prime }+104 y^{\prime \prime \prime }+2740 y^{\prime \prime } = 0 \] The characteristic equation is \[ \lambda ^{4}+104 \lambda ^{3}+2740 \lambda ^{2} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 0\\ \lambda _3 &= -52+6 i\\ \lambda _4 &= -52-6 i \end {align*}

Therefore the homogeneous solution is \[ y_h(x)=c_{2} x +c_{1} +{\mathrm e}^{\left (-52-6 i\right ) x} c_{3} +{\mathrm e}^{\left (-52+6 i\right ) x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= x \\ y_3 &= {\mathrm e}^{\left (-52-6 i\right ) x} \\ y_4 &= {\mathrm e}^{\left (-52+6 i\right ) x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }+104 y^{\prime \prime \prime }+2740 y^{\prime \prime } = 5 \,{\mathrm e}^{-2 x} \cos \left (3 x \right ) \] The particular solution is found using the method of undetermined coeﬃcients. Looking at the RHS of the ode, which is \[ 5 \,{\mathrm e}^{-2 x} \cos \left (3 x \right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{{\mathrm e}^{-2 x} \cos \left (3 x \right ), {\mathrm e}^{-2 x} \sin \left (3 x \right )\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{1, x, {\mathrm e}^{\left (-52-6 i\right ) x}, {\mathrm e}^{\left (-52+6 i\right ) x}\} \] 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. \[ y_p = A_{1} {\mathrm e}^{-2 x} \cos \left (3 x \right )+A_{2} {\mathrm e}^{-2 x} \sin \left (3 x \right ) \] 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 \[ -9035 A_{1} {\mathrm e}^{-2 x} \cos \left (3 x \right )+31824 A_{1} {\mathrm e}^{-2 x} \sin \left (3 x \right )-9035 A_{2} {\mathrm e}^{-2 x} \sin \left (3 x \right )-31824 A_{2} {\mathrm e}^{-2 x} \cos \left (3 x \right ) = 5 \,{\mathrm e}^{-2 x} \cos \left (3 x \right ) \] Solving for the unknowns by comparing coeﬃcients results in \[ \left [A_{1} = -{\frac {3475}{84184477}}, A_{2} = -{\frac {12240}{84184477}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = -\frac {3475 \,{\mathrm e}^{-2 x} \cos \left (3 x \right )}{84184477}-\frac {12240 \,{\mathrm e}^{-2 x} \sin \left (3 x \right )}{84184477} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{2} x +c_{1} +{\mathrm e}^{\left (-52-6 i\right ) x} c_{3} +{\mathrm e}^{\left (-52+6 i\right ) x} c_{4}\right ) + \left (-\frac {3475 \,{\mathrm e}^{-2 x} \cos \left (3 x \right )}{84184477}-\frac {12240 \,{\mathrm e}^{-2 x} \sin \left (3 x \right )}{84184477}\right ) \\ \end{align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} x +c_{1} +{\mathrm e}^{\left (-52-6 i\right ) x} c_{3} +{\mathrm e}^{\left (-52+6 i\right ) x} c_{4} -\frac {3475 \,{\mathrm e}^{-2 x} \cos \left (3 x \right )}{84184477}-\frac {12240 \,{\mathrm e}^{-2 x} \sin \left (3 x \right )}{84184477} \\ \end{align*}

Veriﬁcation of solutions

\[ y = c_{2} x +c_{1} +{\mathrm e}^{\left (-52-6 i\right ) x} c_{3} +{\mathrm e}^{\left (-52+6 i\right ) x} c_{4} -\frac {3475 \,{\mathrm e}^{-2 x} \cos \left (3 x \right )}{84184477}-\frac {12240 \,{\mathrm e}^{-2 x} \sin \left (3 x \right )}{84184477} \] Veriﬁed OK.

Maple trace

`Methods for high order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 4; linear nonhomogeneous with symmetry [0,1] 
-> Calling odsolve with the ODE`, diff(diff(_b(_a), _a), _a) = 5*exp(-2*_a)*cos(3*_a)-104*(diff(_b(_a), _a))-2740*_b(_a), _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: 4; linear nonhomogeneous with symmetry [0,1] successful`

\[ y \left (x \right ) = \frac {\left (\left (667 c_{1} +156 c_{2} \right ) \cos \left (6 x \right )-156 \left (c_{1} -\frac {667 c_{2}}{156}\right ) \sin \left (6 x \right )\right ) {\mathrm e}^{-52 x}}{1876900}+\frac {5 \left (-695 \cos \left (3 x \right )-2448 \sin \left (3 x \right )\right ) {\mathrm e}^{-2 x}}{84184477}+c_{3} x +c_{4} \]

\[ y(x)\to -\frac {12240 e^{-2 x} \sin (3 x)}{84184477}-\frac {3475 e^{-2 x} \cos (3 x)}{84184477}+c_4 x+\frac {(156 c_1+667 c_2) e^{-52 x} \cos (6 x)}{1876900}+\frac {(667 c_1-156 c_2) e^{-52 x} \sin (6 x)}{1876900}+c_3 \]

7.16 problem Problem 41