problem 1(c)

Internal problem ID [5993]
Internal ﬁle name [OUTPUT/5241_Sunday_June_05_2022_03_28_16_PM_76447078/index.tex]

Book: An introduction to Ordinary Diﬀerential Equations. Earl A. Coddington. Dover. NY 1961
Section: Chapter 2. Linear equations with constant coeﬃcients. Page 89
Problem number: 1(c).
ODE order: 4.
ODE degree: 1.

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

\[ \boxed {y^{\prime \prime \prime \prime }+16 y=\cos \left (x \right )} \] 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 \[ y^{\prime \prime \prime \prime }+16 y = 0 \] The characteristic equation is \[ \lambda ^{4}+16 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= \sqrt {2}+i \sqrt {2}\\ \lambda _2 &= -\sqrt {2}+i \sqrt {2}\\ \lambda _3 &= -\sqrt {2}-i \sqrt {2}\\ \lambda _4 &= \sqrt {2}-i \sqrt {2} \end {align*}

Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (-\sqrt {2}+i \sqrt {2}\right ) x} c_{1} +{\mathrm e}^{\left (\sqrt {2}+i \sqrt {2}\right ) x} c_{2} +{\mathrm e}^{\left (\sqrt {2}-i \sqrt {2}\right ) x} c_{3} +{\mathrm e}^{\left (-\sqrt {2}-i \sqrt {2}\right ) x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{\left (-\sqrt {2}+i \sqrt {2}\right ) x} \\ y_2 &= {\mathrm e}^{\left (\sqrt {2}+i \sqrt {2}\right ) x} \\ y_3 &= {\mathrm e}^{\left (\sqrt {2}-i \sqrt {2}\right ) x} \\ y_4 &= {\mathrm e}^{\left (-\sqrt {2}-i \sqrt {2}\right ) x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }+16 y = \cos \left (x \right ) \] The particular solution is found using the method of undetermined coeﬃcients. Looking at the RHS of the ode, which is \[ \cos \left (x \right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{\cos \left (x \right ), \sin \left (x \right )\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \left \{{\mathrm e}^{\left (-\sqrt {2}-i \sqrt {2}\right ) x}, {\mathrm e}^{\left (-\sqrt {2}+i \sqrt {2}\right ) x}, {\mathrm e}^{\left (\sqrt {2}-i \sqrt {2}\right ) x}, {\mathrm e}^{\left (\sqrt {2}+i \sqrt {2}\right ) x}\right \} \] 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. \[ y_p = A_{1} \cos \left (x \right )+A_{2} \sin \left (x \right ) \] 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 \[ 17 A_{1} \cos \left (x \right )+17 A_{2} \sin \left (x \right ) = \cos \left (x \right ) \] Solving for the unknowns by comparing coeﬃcients results in \[ \left [A_{1} = {\frac {1}{17}}, A_{2} = 0\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {\cos \left (x \right )}{17} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{\left (-\sqrt {2}+i \sqrt {2}\right ) x} c_{1} +{\mathrm e}^{\left (\sqrt {2}+i \sqrt {2}\right ) x} c_{2} +{\mathrm e}^{\left (\sqrt {2}-i \sqrt {2}\right ) x} c_{3} +{\mathrm e}^{\left (-\sqrt {2}-i \sqrt {2}\right ) x} c_{4}\right ) + \left (\frac {\cos \left (x \right )}{17}\right ) \\ \end{align*} Which simpliﬁes to \[ y = {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} c_{1} +{\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} c_{2} +{\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} c_{3} +{\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} c_{4} +\frac {\cos \left (x \right )}{17} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} c_{1} +{\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} c_{2} +{\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} c_{3} +{\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} c_{4} +\frac {\cos \left (x \right )}{17} \\ \end{align*}

Veriﬁcation of solutions

\[ y = {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} c_{1} +{\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} c_{2} +{\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} c_{3} +{\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} c_{4} +\frac {\cos \left (x \right )}{17} \] 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`

\[ y \left (x \right ) = c_{4} {\mathrm e}^{-\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+c_{2} {\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+c_{3} {\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )+c_{1} {\mathrm e}^{\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )+\frac {\cos \left (x \right )}{17} \]

\[ y(x)\to \frac {\cos (x)}{17}+e^{-\sqrt {2} x} \left (\left (c_1 e^{2 \sqrt {2} x}+c_2\right ) \cos \left (\sqrt {2} x\right )+\left (c_4 e^{2 \sqrt {2} x}+c_3\right ) \sin \left (\sqrt {2} x\right )\right ) \]

10.3 problem 1(c)