Internal problem ID [15275]
Internal file name [OUTPUT/15276_Wednesday_May_08_2024_03_54_39_PM_20973667/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 15.3 Nonhomogeneous linear equations with
constant coefficients. Trial and error method. Exercises page 132
Problem number: 506.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }-2 n^{2} y^{\prime \prime }+n^{4} y=\cos \left (n x +\alpha \right )} \] 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 &= n\\ \lambda _2 &= n\\ \lambda _3 &= -n\\ \lambda _4 &= -n \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{-n x} c_{1} +x \,{\mathrm e}^{-n x} c_{2} +{\mathrm e}^{n x} c_{3} +x \,{\mathrm e}^{n x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{-n x} \\ y_2 &= x \,{\mathrm e}^{-n x} \\ y_3 &= {\mathrm e}^{n x} \\ y_4 &= x \,{\mathrm e}^{n x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }-2 n^{2} y^{\prime \prime }+n^{4} y = \cos \left (n x +\alpha \right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ \cos \left (n x +\alpha \right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{\cos \left (n x +\alpha \right ), \sin \left (n x +\alpha \right )\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{x \,{\mathrm e}^{n x}, x \,{\mathrm e}^{-n x}, {\mathrm e}^{n x}, {\mathrm e}^{-n x}\} \] 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. \[ y_p = A_{1} \cos \left (n x +\alpha \right )+A_{2} \sin \left (n x +\alpha \right ) \] 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 \[ A_{1} n^{4} \cos \left (n x +\alpha \right )+A_{2} n^{4} \sin \left (n x +\alpha \right )-2 n^{2} \left (-A_{1} n^{2} \cos \left (n x +\alpha \right )-A_{2} n^{2} \sin \left (n x +\alpha \right )\right )+n^{4} \left (A_{1} \cos \left (n x +\alpha \right )+A_{2} \sin \left (n x +\alpha \right )\right ) = \cos \left (n x +\alpha \right ) \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = \frac {1}{4 n^{4}}, A_{2} = 0\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {\cos \left (n x +\alpha \right )}{4 n^{4}} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{-n x} c_{1} +x \,{\mathrm e}^{-n x} c_{2} +{\mathrm e}^{n x} c_{3} +x \,{\mathrm e}^{n x} c_{4}\right ) + \left (\frac {\cos \left (n x +\alpha \right )}{4 n^{4}}\right ) \\ \end{align*} Which simplifies to \[ y = \left (c_{2} x +c_{1} \right ) {\mathrm e}^{-n x}+{\mathrm e}^{n x} \left (c_{4} x +c_{3} \right )+\frac {\cos \left (n x +\alpha \right )}{4 n^{4}} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \left (c_{2} x +c_{1} \right ) {\mathrm e}^{-n x}+{\mathrm e}^{n x} \left (c_{4} x +c_{3} \right )+\frac {\cos \left (n x +\alpha \right )}{4 n^{4}} \\ \end{align*}
Verification of solutions
\[ y = \left (c_{2} x +c_{1} \right ) {\mathrm e}^{-n x}+{\mathrm e}^{n x} \left (c_{4} x +c_{3} \right )+\frac {\cos \left (n x +\alpha \right )}{4 n^{4}} \] 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: 0.016 (sec). Leaf size: 49
dsolve(diff(y(x),x$4)-2*n^2*diff(y(x),x$2)+n^4*y(x)=cos(n*x+alpha),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\cos \left (n x +\alpha \right )+\left (4 c_{4} x +4 c_{2} \right ) n^{4} {\mathrm e}^{-n x}+\left (4 c_{3} x +4 c_{1} \right ) n^{4} {\mathrm e}^{n x}}{4 n^{4}} \]
✓ Solution by Mathematica
Time used: 0.234 (sec). Leaf size: 52
DSolve[y''''[x]-2*n^2*y''[x]+n^4*y[x]==Cos[n*x+\[Alpha]],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\cos (\alpha +n x)}{4 n^4}+e^{-n x} \left (c_3 e^{2 n x}+c_4 x e^{2 n x}+c_2 x+c_1\right ) \]