11.20 problem 20

Internal problem ID [2159]
Internal file name [OUTPUT/2159_Monday_February_26_2024_09_17_56_AM_44492199/index.tex]

Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section: Exercise 19, page 86
Problem number: 20.
ODE order: 4.
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

[[_high_order, _linear, _nonhomogeneous]]

\[ \boxed {y^{\prime \prime \prime \prime }+2 n^{2} y^{\prime \prime }+n^{4} y=\sin \left (k x \right )} \] 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 \[ y^{\prime \prime \prime \prime }+2 n^{2} y^{\prime \prime }+n^{4} y = 0 \] The characteristic equation is \[ \lambda ^{4}+2 \lambda ^{2} n^{2}+n^{4} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= i n\\ \lambda _2 &= -i n\\ \lambda _3 &= i n\\ \lambda _4 &= -i n \end {align*}

Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{i n x} c_{1} +x \,{\mathrm e}^{i n x} c_{2} +{\mathrm e}^{-i n x} c_{3} +x \,{\mathrm e}^{-i n x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{i n x} \\ y_2 &= x \,{\mathrm e}^{i n x} \\ y_3 &= {\mathrm e}^{-i n x} \\ y_4 &= x \,{\mathrm e}^{-i n x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }+2 n^{2} y^{\prime \prime }+n^{4} y = \sin \left (k x \right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ \sin \left (k x \right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{\cos \left (k x \right ), \sin \left (k x \right )\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{x \,{\mathrm e}^{i n x}, x \,{\mathrm e}^{-i n x}, {\mathrm e}^{i n x}, {\mathrm e}^{-i n x}\} \] 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. \[ y_p = A_{1} \cos \left (k x \right )+A_{2} \sin \left (k x \right ) \] 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 \[ A_{1} k^{4} \cos \left (k x \right )+A_{2} k^{4} \sin \left (k x \right )+2 n^{2} \left (-A_{1} k^{2} \cos \left (k x \right )-A_{2} k^{2} \sin \left (k x \right )\right )+n^{4} \left (A_{1} \cos \left (k x \right )+A_{2} \sin \left (k x \right )\right ) = \sin \left (k x \right ) \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = 0, A_{2} = \frac {1}{\left (k -n \right )^{2} \left (k +n \right )^{2}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {\sin \left (k x \right )}{\left (k -n \right )^{2} \left (k +n \right )^{2}} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{i n x} c_{1} +x \,{\mathrm e}^{i n x} c_{2} +{\mathrm e}^{-i n x} c_{3} +x \,{\mathrm e}^{-i n x} c_{4}\right ) + \left (\frac {\sin \left (k x \right )}{\left (k -n \right )^{2} \left (k +n \right )^{2}}\right ) \\ \end{align*} Which simplifies to \[ y = \left (c_{2} x +c_{1} \right ) {\mathrm e}^{i n x}+\left (c_{4} x +c_{3} \right ) {\mathrm e}^{-i n x}+\frac {\sin \left (k x \right )}{\left (k -n \right )^{2} \left (k +n \right )^{2}} \]

Summary

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

Verification of solutions

\[ y = \left (c_{2} x +c_{1} \right ) {\mathrm e}^{i n x}+\left (c_{4} x +c_{3} \right ) {\mathrm e}^{-i n x}+\frac {\sin \left (k x \right )}{\left (k -n \right )^{2} \left (k +n \right )^{2}} \] 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] 
trying high order linear exact nonhomogeneous 
trying differential order: 4; missing the dependent variable 
checking if the LODE has constant coefficients 
<- constant coefficients successful`
 

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 48

dsolve(diff(y(x),x$4)+2*n^2*diff(y(x),x$2)+n^4*y(x)=sin(k*x),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\sin \left (k x \right )}{\left (k -n \right )^{2} \left (k +n \right )^{2}}+c_{1} \cos \left (n x \right )+c_{2} \sin \left (n x \right )+c_{3} \cos \left (n x \right ) x +c_{4} \sin \left (n x \right ) x \]

✓ Solution by Mathematica

Time used: 0.457 (sec). Leaf size: 69

DSolve[y''''[x]+2*n^2*y''[x]+n^4*y[x]==Sin[k*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {(c_2 x+c_1) \left (k^2-n^2\right )^2 \cos (n x)+(c_4 x+c_3) \left (k^2-n^2\right )^2 \sin (n x)+\sin (k x)}{(k-n)^2 (k+n)^2} \]