Internal problem ID [12770]
Internal file name [OUTPUT/11423_Friday_November_03_2023_06_32_48_AM_73742763/index.tex]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 4. N-th Order Linear Differential Equations. Exercises 4.3, page 210
Problem number: 17.
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]]
\[ \boxed {y^{\prime \prime \prime }+\left (-3-4 i\right ) y^{\prime \prime }+\left (-4+12 i\right ) y^{\prime }+12 y=0} \] The characteristic equation is \[ \lambda ^{3}-4 i \lambda ^{2}-3 \lambda ^{2}+12 i \lambda -4 \lambda +12 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 3\\ \lambda _2 &= 2 i\\ \lambda _3 &= 2 i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{3 x}+{\mathrm e}^{2 i x} c_{2} +x \,{\mathrm e}^{2 i x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{3 x}\\ y_2 &= {\mathrm e}^{2 i x}\\ y_3 &= x \,{\mathrm e}^{2 i x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{3 x}+{\mathrm e}^{2 i x} c_{2} +x \,{\mathrm e}^{2 i x} c_{3} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{3 x}+{\mathrm e}^{2 i x} c_{2} +x \,{\mathrm e}^{2 i x} c_{3} \] 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: 0.015 (sec). Leaf size: 22
dsolve(diff(y(x),x$3)-(3+4*I)*diff(y(x),x$2)-(4-12*I)*diff(y(x),x)+12*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{3} x +c_{2} \right ) {\mathrm e}^{2 i x}+c_{1} {\mathrm e}^{3 x} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 29
DSolve[y'''[x]-(3+4*I)*y''[x]-(4-12*I)*y'[x]+12*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{2 i x} (c_2 x+c_1)+c_3 e^{3 x} \]