Internal problem ID [12771]
Internal file name [OUTPUT/11424_Friday_November_03_2023_06_32_48_AM_49111732/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: 18.
ODE order: 4.
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
[[_high_order, _missing_x]]
\[ \boxed {y^{\prime \prime \prime \prime }+\left (-3-i\right ) y^{\prime \prime \prime }+\left (4+3 i\right ) y^{\prime \prime }=0} \] The characteristic equation is \[ \lambda ^{4}-i \lambda ^{3}-3 \lambda ^{3}+3 i \lambda ^{2}+4 \lambda ^{2} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 0\\ \lambda _3 &= 1+2 i\\ \lambda _4 &= 2-i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{2} x +c_{1} +{\mathrm e}^{\left (1+2 i\right ) x} c_{3} +{\mathrm e}^{\left (2-i\right ) x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= 1\\ y_2 &= x\\ y_3 &= {\mathrm e}^{\left (1+2 i\right ) x}\\ y_4 &= {\mathrm e}^{\left (2-i\right ) x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} x +c_{1} +{\mathrm e}^{\left (1+2 i\right ) x} c_{3} +{\mathrm e}^{\left (2-i\right ) x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = c_{2} x +c_{1} +{\mathrm e}^{\left (1+2 i\right ) x} c_{3} +{\mathrm e}^{\left (2-i\right ) x} c_{4} \] 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: 0.015 (sec). Leaf size: 25
dsolve(diff(y(x),x$4)-(3+I)*diff(y(x),x$3)+(4+3*I)*diff(y(x),x$2)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{\left (1+2 i\right ) x}+c_{2} {\mathrm e}^{\left (2-i\right ) x}+c_{3} +c_{4} x \]
✓ Solution by Mathematica
Time used: 0.156 (sec). Leaf size: 46
DSolve[y''''[x]-(3+I)*y'''[x]+(4+3*I)*y''[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (-\frac {3}{25}-\frac {4 i}{25}\right ) c_1 e^{(1+2 i) x}+\left (\frac {3}{25}+\frac {4 i}{25}\right ) c_2 e^{(2-i) x}+c_4 x+c_3 \]