Internal problem ID [5367]
Internal file name [OUTPUT/4858_Sunday_February_04_2024_12_46_32_AM_13284602/index.tex]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 13. Homogeneous Linear equations with constant coefficients. Supplemetary
problems. Page 86
Problem number: 25.
ODE order: 6.
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]]
\[ \boxed {y^{\left (6\right )}+9 y^{\prime \prime \prime \prime }+24 y^{\prime \prime }+16 y=0} \] The characteristic equation is \[ \lambda ^{6}+9 \lambda ^{4}+24 \lambda ^{2}+16 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= i\\ \lambda _2 &= -i\\ \lambda _3 &= 2 i\\ \lambda _4 &= -2 i\\ \lambda _5 &= 2 i\\ \lambda _6 &= -2 i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{i x} c_{1} +{\mathrm e}^{-i x} c_{2} +{\mathrm e}^{2 i x} c_{3} +x \,{\mathrm e}^{2 i x} c_{4} +{\mathrm e}^{-2 i x} c_{5} +x \,{\mathrm e}^{-2 i x} c_{6} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{i x}\\ y_2 &= {\mathrm e}^{-i x}\\ y_3 &= {\mathrm e}^{2 i x}\\ y_4 &= x \,{\mathrm e}^{2 i x}\\ y_5 &= {\mathrm e}^{-2 i x}\\ y_6 &= x \,{\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}^{-i x} c_{2} +{\mathrm e}^{2 i x} c_{3} +x \,{\mathrm e}^{2 i x} c_{4} +{\mathrm e}^{-2 i x} c_{5} +x \,{\mathrm e}^{-2 i x} c_{6} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{i x} c_{1} +{\mathrm e}^{-i x} c_{2} +{\mathrm e}^{2 i x} c_{3} +x \,{\mathrm e}^{2 i x} c_{4} +{\mathrm e}^{-2 i x} c_{5} +x \,{\mathrm e}^{-2 i x} c_{6} \] 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: 33
dsolve(diff(y(x),x$6)+9*diff(y(x),x$4)+24*diff(y(x),x$2)+16*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{6} x +c_{4} \right ) \cos \left (2 x \right )+\left (c_{5} x +c_{3} \right ) \sin \left (2 x \right )+c_{1} \sin \left (x \right )+\cos \left (x \right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 40
DSolve[y''''''[x]+9*y''''[x]+24*y''[x]+16*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (c_2 x+c_1) \cos (2 x)+c_6 \sin (x)+\cos (x) (2 (c_4 x+c_3) \sin (x)+c_5) \]