Internal problem ID [5979]
Internal file name [OUTPUT/5227_Sunday_June_05_2022_03_27_57_PM_31923992/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(i).
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]]
\[ \boxed {y^{\prime \prime \prime }-3 i y^{\prime \prime }-3 y^{\prime }+i y=0} \] The characteristic equation is \[ \lambda ^{3}-3 i \lambda ^{2}-3 \lambda +i = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= i\\ \lambda _2 &= i\\ \lambda _3 &= i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{i x} c_{1} +x \,{\mathrm e}^{i x} c_{2} +x^{2} {\mathrm e}^{i x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{i x}\\ y_2 &= x \,{\mathrm e}^{i x}\\ y_3 &= x^{2} {\mathrm e}^{i x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{i x} c_{1} +x \,{\mathrm e}^{i x} c_{2} +x^{2} {\mathrm e}^{i x} c_{3} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{i x} c_{1} +x \,{\mathrm e}^{i x} c_{2} +x^{2} {\mathrm e}^{i x} c_{3} \] 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: 0.0 (sec). Leaf size: 20
dsolve(diff(y(x),x$3)-3*I*diff(y(x),x$2)-3*diff(y(x),x)+I*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{i x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 25
DSolve[y'''[x]-3*I*y''[x]-3*y'[x]+I*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{i x} (x (c_3 x+c_2)+c_1) \]