Internal problem ID [5975]
Internal file name [OUTPUT/5223_Sunday_June_05_2022_03_27_52_PM_23069737/index.tex]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 2. Linear equations with constant coefficients. Page 74
Problem number: 4(d).
ODE order: 3.
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
[[_3rd_order, _missing_x]]
\[ \boxed {y^{\prime \prime \prime }-i y^{\prime \prime }+4 y^{\prime }-4 i y=0} \] The characteristic equation is \[ \lambda ^{3}-i \lambda ^{2}+4 \lambda -4 i = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -2 i\\ \lambda _2 &= 2 i\\ \lambda _3 &= i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{i x} c_{1} +{\mathrm e}^{2 i x} c_{2} +{\mathrm e}^{-2 i x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{i x}\\ y_2 &= {\mathrm e}^{2 i x}\\ y_3 &= {\mathrm e}^{-2 i x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{i x} c_{1} +{\mathrm e}^{2 i x} c_{2} +{\mathrm e}^{-2 i x} c_{3} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{i x} c_{1} +{\mathrm e}^{2 i x} c_{2} +{\mathrm e}^{-2 i x} c_{3} \] Verified OK.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 26
dsolve(diff(y(x),x$3)-I*diff(y(x),x$2)+4*diff(y(x),x)-4*I*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{i x}+c_{2} {\mathrm e}^{2 i x}+c_{3} {\mathrm e}^{-2 i x} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 36
DSolve[y'''[x]-I*y''[x]+4*y'[x]-4*I*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-2 i x} \left (c_2 e^{4 i x}+c_3 e^{3 i x}+c_1\right ) \]