Internal problem ID [12775]
Internal file name [OUTPUT/11428_Friday_November_03_2023_06_32_51_AM_25526768/index.tex]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 4. N-th Order Linear Differential Equations. Exercises 4.4, page 218
Problem number: 3.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y=3+\cos \left (2 x \right )} \] 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 \[ y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y = 0 \] The characteristic equation is \[ \lambda ^{4}+2 \lambda ^{2}+1 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= i\\ \lambda _2 &= -i\\ \lambda _3 &= i\\ \lambda _4 &= -i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{i x} c_{1} +x \,{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-i x} c_{3} +x \,{\mathrm e}^{-i x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{i x} \\ y_2 &= x \,{\mathrm e}^{i x} \\ y_3 &= {\mathrm e}^{-i x} \\ y_4 &= x \,{\mathrm e}^{-i x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y = 3+\cos \left (2 x \right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ 3+\cos \left (2 x \right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{1\}, \{\cos \left (2 x \right ), \sin \left (2 x \right )\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{x \,{\mathrm e}^{i x}, x \,{\mathrm e}^{-i x}, {\mathrm e}^{i x}, {\mathrm e}^{-i x}\} \] 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. \[ y_p = A_{1}+A_{2} \cos \left (2 x \right )+A_{3} \sin \left (2 x \right ) \] The unknowns \(\{A_{1}, A_{2}, A_{3}\}\) 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 \[ 9 A_{2} \cos \left (2 x \right )+9 A_{3} \sin \left (2 x \right )+A_{1} = 3+\cos \left (2 x \right ) \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = 3, A_{2} = {\frac {1}{9}}, A_{3} = 0\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = 3+\frac {\cos \left (2 x \right )}{9} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{i x} c_{1} +x \,{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-i x} c_{3} +x \,{\mathrm e}^{-i x} c_{4}\right ) + \left (3+\frac {\cos \left (2 x \right )}{9}\right ) \\ \end{align*} Which simplifies to \[ y = \left (c_{2} x +c_{1} \right ) {\mathrm e}^{i x}+\left (c_{4} x +c_{3} \right ) {\mathrm e}^{-i x}+3+\frac {\cos \left (2 x \right )}{9} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \left (c_{2} x +c_{1} \right ) {\mathrm e}^{i x}+\left (c_{4} x +c_{3} \right ) {\mathrm e}^{-i x}+3+\frac {\cos \left (2 x \right )}{9} \\ \end{align*}
Verification of solutions
\[ y = \left (c_{2} x +c_{1} \right ) {\mathrm e}^{i x}+\left (c_{4} x +c_{3} \right ) {\mathrm e}^{-i x}+3+\frac {\cos \left (2 x \right )}{9} \] 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: 28
dsolve(diff(y(x),x$4)+2*diff(y(x),x$2)+y(x)=3+cos(2*x),y(x), singsol=all)
\[ y \left (x \right ) = 3+\frac {\cos \left (2 x \right )}{9}+\left (c_{4} x +c_{1} \right ) \cos \left (x \right )+\left (c_{3} x +c_{2} \right ) \sin \left (x \right ) \]
✓ Solution by Mathematica
Time used: 0.199 (sec). Leaf size: 36
DSolve[y''''[x]+2*y''[x]+y[x]==3+Cos[2*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{9} \cos (2 x)+(c_2 x+c_1) \cos (x)+c_3 \sin (x)+c_4 x \sin (x)+3 \]