Internal problem ID [11772]
Internal file name [OUTPUT/11782_Thursday_April_11_2024_08_49_37_PM_10735052/index.tex]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 4, Section 4.2. The homogeneous linear equation with constant coefficients.
Exercises page 135
Problem number: 45.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+6 y^{\prime \prime }+2 y^{\prime }+5 y=0} \] The characteristic equation is \[ \lambda ^{4}+2 \lambda ^{3}+6 \lambda ^{2}+2 \lambda +5 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= i\\ \lambda _2 &= -i\\ \lambda _3 &= -1-2 i\\ \lambda _4 &= -1+2 i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (-1-2 i\right ) x} c_{1} +{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-i x} c_{3} +{\mathrm e}^{\left (-1+2 i\right ) x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{\left (-1-2 i\right ) x}\\ y_2 &= {\mathrm e}^{i x}\\ y_3 &= {\mathrm e}^{-i x}\\ y_4 &= {\mathrm e}^{\left (-1+2 i\right ) x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\left (-1-2 i\right ) x} c_{1} +{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-i x} c_{3} +{\mathrm e}^{\left (-1+2 i\right ) x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{\left (-1-2 i\right ) x} c_{1} +{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-i x} c_{3} +{\mathrm e}^{\left (-1+2 i\right ) x} c_{4} \] 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: 0.0 (sec). Leaf size: 36
dsolve(diff(y(x),x$4)+2*diff(y(x),x$3)+6*diff(y(x),x$2)+2*diff(y(x),x)+5*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (2 c_{3} \cos \left (x \right ) \sin \left (x \right )+2 c_{4} \cos \left (x \right )^{2}-c_{4} \right ) {\mathrm e}^{-x}+c_{1} \sin \left (x \right )+c_{2} \cos \left (x \right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 40
DSolve[y''''[x]+2*y'''[x]+6*y''[x]+2*y'[x]+5*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_3 \cos (x)+e^{-x} \left (c_2 \cos (2 x)+c_4 e^x \sin (x)+c_1 \sin (2 x)\right ) \]