Internal problem ID [13636]
Internal file name [OUTPUT/12808_Saturday_February_17_2024_08_44_55_AM_37514711/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 19. Arbitrary Homogeneous linear equations with constant coefficients.
Additional exercises page 369
Problem number: 19.4 (e).
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]]
\[ \boxed {y^{\left (6\right )}-3 y^{\prime \prime \prime \prime }+3 y^{\prime \prime }-y=0} \] The characteristic equation is \[ \lambda ^{6}-3 \lambda ^{4}+3 \lambda ^{2}-1 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 1\\ \lambda _2 &= -1\\ \lambda _3 &= 1\\ \lambda _4 &= -1\\ \lambda _5 &= 1\\ \lambda _6 &= -1 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{-x} c_{1} +x \,{\mathrm e}^{-x} c_{2} +x^{2} {\mathrm e}^{-x} c_{3} +c_{4} {\mathrm e}^{x}+x \,{\mathrm e}^{x} c_{5} +x^{2} {\mathrm e}^{x} c_{6} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-x}\\ y_2 &= x \,{\mathrm e}^{-x}\\ y_3 &= {\mathrm e}^{-x} x^{2}\\ y_4 &= {\mathrm e}^{x}\\ y_5 &= x \,{\mathrm e}^{x}\\ y_6 &= {\mathrm e}^{x} x^{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-x} c_{1} +x \,{\mathrm e}^{-x} c_{2} +x^{2} {\mathrm e}^{-x} c_{3} +c_{4} {\mathrm e}^{x}+x \,{\mathrm e}^{x} c_{5} +x^{2} {\mathrm e}^{x} c_{6} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{-x} c_{1} +x \,{\mathrm e}^{-x} c_{2} +x^{2} {\mathrm e}^{-x} c_{3} +c_{4} {\mathrm e}^{x}+x \,{\mathrm e}^{x} c_{5} +x^{2} {\mathrm e}^{x} c_{6} \] 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)-3*diff(y(x),x$4)+3*diff(y(x),x$2)-y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (x^{2} c_{3} +c_{2} x +c_{1} \right ) {\mathrm e}^{-x}+{\mathrm e}^{x} \left (c_{6} x^{2}+c_{5} x +c_{4} \right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 50
DSolve[y''''''[x]-3*y''''[x]+3*y''[x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-x} \left (x^2 \left (c_6 e^{2 x}+c_3\right )+x \left (c_5 e^{2 x}+c_2\right )+c_4 e^{2 x}+c_1\right ) \]