Internal problem ID [13641]
Internal file name [OUTPUT/12813_Saturday_February_17_2024_08_44_56_AM_86667132/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 (j).
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]]
\[ \boxed {y^{\left (6\right )}+16 y^{\prime \prime \prime }+64 y=0} \] The characteristic equation is \[ \lambda ^{6}+16 \lambda ^{3}+64 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -2\\ \lambda _2 &= 1-i \sqrt {3}\\ \lambda _3 &= 1+i \sqrt {3}\\ \lambda _4 &= -2\\ \lambda _5 &= 1-i \sqrt {3}\\ \lambda _6 &= 1+i \sqrt {3} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{-2 x}+x \,{\mathrm e}^{-2 x} c_{2} +{\mathrm e}^{\left (1+i \sqrt {3}\right ) x} c_{3} +x \,{\mathrm e}^{\left (1+i \sqrt {3}\right ) x} c_{4} +{\mathrm e}^{\left (1-i \sqrt {3}\right ) x} c_{5} +x \,{\mathrm e}^{\left (1-i \sqrt {3}\right ) x} c_{6} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-2 x}\\ y_2 &= {\mathrm e}^{-2 x} x\\ y_3 &= {\mathrm e}^{\left (1+i \sqrt {3}\right ) x}\\ y_4 &= x \,{\mathrm e}^{\left (1+i \sqrt {3}\right ) x}\\ y_5 &= {\mathrm e}^{\left (1-i \sqrt {3}\right ) x}\\ y_6 &= x \,{\mathrm e}^{\left (1-i \sqrt {3}\right ) x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{-2 x}+x \,{\mathrm e}^{-2 x} c_{2} +{\mathrm e}^{\left (1+i \sqrt {3}\right ) x} c_{3} +x \,{\mathrm e}^{\left (1+i \sqrt {3}\right ) x} c_{4} +{\mathrm e}^{\left (1-i \sqrt {3}\right ) x} c_{5} +x \,{\mathrm e}^{\left (1-i \sqrt {3}\right ) x} c_{6} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{-2 x}+x \,{\mathrm e}^{-2 x} c_{2} +{\mathrm e}^{\left (1+i \sqrt {3}\right ) x} c_{3} +x \,{\mathrm e}^{\left (1+i \sqrt {3}\right ) x} c_{4} +{\mathrm e}^{\left (1-i \sqrt {3}\right ) x} c_{5} +x \,{\mathrm e}^{\left (1-i \sqrt {3}\right ) x} c_{6} \] 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: 46
dsolve(diff(y(x),x$6)+16*diff(y(x),x$3)+64*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left ({\mathrm e}^{3 x} \left (c_{6} x +c_{4} \right ) \cos \left (\sqrt {3}\, x \right )+{\mathrm e}^{3 x} \left (c_{5} x +c_{3} \right ) \sin \left (\sqrt {3}\, x \right )+c_{2} x +c_{1} \right ) {\mathrm e}^{-2 x} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 60
DSolve[y''''''[x]+16*y'''[x]+64*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-2 x} \left (c_6 x+e^{3 x} (c_4 x+c_3) \cos \left (\sqrt {3} x\right )+e^{3 x} (c_2 x+c_1) \sin \left (\sqrt {3} x\right )+c_5\right ) \]