Internal problem ID [13637]
Internal file name [OUTPUT/12809_Saturday_February_17_2024_08_44_55_AM_234803/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 (f).
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]]
\[ \boxed {y^{\left (6\right )}-2 y^{\prime \prime \prime }+y=0} \] The characteristic equation is \[ \lambda ^{6}-2 \lambda ^{3}+1 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 1\\ \lambda _2 &= -\frac {1}{2}-\frac {i \sqrt {3}}{2}\\ \lambda _3 &= -\frac {1}{2}+\frac {i \sqrt {3}}{2}\\ \lambda _4 &= 1\\ \lambda _5 &= -\frac {1}{2}-\frac {i \sqrt {3}}{2}\\ \lambda _6 &= -\frac {1}{2}+\frac {i \sqrt {3}}{2} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{x}+c_{2} x \,{\mathrm e}^{x}+{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x} c_{3} +x \,{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x} c_{4} +{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x} c_{5} +x \,{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x} c_{6} \] 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}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x}\\ y_4 &= x \,{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x}\\ y_5 &= {\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x}\\ y_6 &= x \,{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{x}+c_{2} x \,{\mathrm e}^{x}+{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x} c_{3} +x \,{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x} c_{4} +{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x} c_{5} +x \,{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x} c_{6} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{x}+c_{2} x \,{\mathrm e}^{x}+{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x} c_{3} +x \,{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x} c_{4} +{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x} c_{5} +x \,{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x} c_{6} \] 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: 47
dsolve(diff(y(x),x$6)-2*diff(y(x),x$3)+y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {x}{2}} \left (c_{6} x +c_{4} \right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )+{\mathrm e}^{-\frac {x}{2}} \left (c_{5} x +c_{3} \right ) \sin \left (\frac {\sqrt {3}\, x}{2}\right )+{\mathrm e}^{x} \left (c_{2} x +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 67
DSolve[y''''''[x]-2*y'''[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-x/2} \left (e^{3 x/2} (c_6 x+c_5)+(c_4 x+c_3) \cos \left (\frac {\sqrt {3} x}{2}\right )+(c_2 x+c_1) \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \]