Internal problem ID [5990]
Internal file name [OUTPUT/5238_Sunday_June_05_2022_03_28_11_PM_31224304/index.tex]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 2. Linear equations with constant coefficients. Page 83
Problem number: 5(b).
ODE order: 4.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_high_order, _missing_x]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime \prime \prime }-k^{4} y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0, y \left (1\right ) = 0, y^{\prime }\left (1\right ) = 0] \end {align*}
The characteristic equation is \[ -k^{4}+\lambda ^{4} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= k\\ \lambda _2 &= -k\\ \lambda _3 &= i k\\ \lambda _4 &= -i k \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{i k x} c_{1} +{\mathrm e}^{k x} c_{2} +{\mathrm e}^{-i k x} c_{3} +{\mathrm e}^{-k x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{i k x}\\ y_2 &= {\mathrm e}^{k x}\\ y_3 &= {\mathrm e}^{-i k x}\\ y_4 &= {\mathrm e}^{-k x} \end {align*}
Initial conditions are used to solve for the constants of integration.
Looking at the above solution \begin {align*} y = {\mathrm e}^{i k x} c_{1} +{\mathrm e}^{k x} c_{2} +{\mathrm e}^{-i k x} c_{3} +{\mathrm e}^{-k x} c_{4} \tag {1} \end {align*}
Initial conditions are now substituted in the above solution. This will generate the required equations to solve for the integration constants. substituting \(y = 0\) and \(x = 1\) in the above gives \begin {align*} 0 = {\mathrm e}^{i k} c_{1} +{\mathrm e}^{k} c_{2} +{\mathrm e}^{-i k} c_{3} +{\mathrm e}^{-k} c_{4}\tag {1A} \end {align*}
substituting \(y = 0\) and \(x = 0\) in the above gives \begin {align*} 0 = c_{1} +c_{2} +c_{3} +c_{4}\tag {2A} \end {align*}
Taking derivative of the solution gives \begin {align*} y^{\prime } = i k \,{\mathrm e}^{i k x} c_{1} +k \,{\mathrm e}^{k x} c_{2} -i k \,{\mathrm e}^{-i k x} c_{3} -k \,{\mathrm e}^{-k x} c_{4} \end {align*}
substituting \(y^{\prime } = 0\) and \(x = 1\) in the above gives \begin {align*} 0 = \left (i {\mathrm e}^{i k} c_{1} +{\mathrm e}^{k} c_{2} -i {\mathrm e}^{-i k} c_{3} -{\mathrm e}^{-k} c_{4} \right ) k\tag {3A} \end {align*}
Taking derivative of the solution gives \begin {align*} y^{\prime } = i k \,{\mathrm e}^{i k x} c_{1} +k \,{\mathrm e}^{k x} c_{2} -i k \,{\mathrm e}^{-i k x} c_{3} -k \,{\mathrm e}^{-k x} c_{4} \end {align*}
substituting \(y^{\prime } = 0\) and \(x = 0\) in the above gives \begin {align*} 0 = \left (c_{1} i-c_{3} i+c_{2} -c_{4} \right ) k\tag {4A} \end {align*}
Equations {1A,2A,3A,4A} are now solved for \(\{c_{1}, c_{2}, c_{3}, c_{4}\}\). Solving for the constants gives \begin {align*} c_{1}&=\myadjustbox {0}\\ c_{2}&=\myadjustbox {0}\\ c_{3}&=\myadjustbox {0}\\ c_{4}&=\myadjustbox {0} \end {align*}
Substituting these values back in above solution results in \begin {align*} y = 0 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 0 \\ \end{align*}
Verification of solutions
\[ y = 0 \] Verified OK.
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 5
dsolve([diff(y(x),x$4)-k^4*y(x)=0,y(0) = 0, D(y)(0) = 0, y(1) = 0, D(y)(1) = 0],y(x), singsol=all)
\[ y \left (x \right ) = 0 \]
✓ Solution by Mathematica
Time used: 0.02 (sec). Leaf size: 6
DSolve[{y''''[x]-k^4*y[x]==0,{y[0]==0,y[1]==0,y'[0]==0,y'[1]==0}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 0 \]