Internal problem ID [3253]
Internal file name [OUTPUT/2745_Sunday_June_05_2022_08_39_57_AM_29914027/index.tex]
Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 4. Linear Differential Equations. Page 183
Problem number: 12.
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]]
\[ \boxed {y^{\left (6\right )}-64 y=0} \] The characteristic equation is \[ \lambda ^{6}-64 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 2\\ \lambda _2 &= -2\\ \lambda _3 &= \sqrt {-2-2 i \sqrt {3}}\\ \lambda _4 &= -\sqrt {-2-2 i \sqrt {3}}\\ \lambda _5 &= \sqrt {-2+2 i \sqrt {3}}\\ \lambda _6 &= -\sqrt {-2+2 i \sqrt {3}} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{-2 x} c_{1} +{\mathrm e}^{2 x} c_{2} +{\mathrm e}^{\sqrt {-2+2 i \sqrt {3}}\, x} c_{3} +{\mathrm e}^{-\sqrt {-2+2 i \sqrt {3}}\, x} c_{4} +{\mathrm e}^{-\sqrt {-2-2 i \sqrt {3}}\, x} c_{5} +{\mathrm e}^{\sqrt {-2-2 i \sqrt {3}}\, x} c_{6} \] 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}\\ y_3 &= {\mathrm e}^{\sqrt {-2+2 i \sqrt {3}}\, x}\\ y_4 &= {\mathrm e}^{-\sqrt {-2+2 i \sqrt {3}}\, x}\\ y_5 &= {\mathrm e}^{-\sqrt {-2-2 i \sqrt {3}}\, x}\\ y_6 &= {\mathrm e}^{\sqrt {-2-2 i \sqrt {3}}\, x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-2 x} c_{1} +{\mathrm e}^{2 x} c_{2} +{\mathrm e}^{\sqrt {-2+2 i \sqrt {3}}\, x} c_{3} +{\mathrm e}^{-\sqrt {-2+2 i \sqrt {3}}\, x} c_{4} +{\mathrm e}^{-\sqrt {-2-2 i \sqrt {3}}\, x} c_{5} +{\mathrm e}^{\sqrt {-2-2 i \sqrt {3}}\, x} c_{6} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{-2 x} c_{1} +{\mathrm e}^{2 x} c_{2} +{\mathrm e}^{\sqrt {-2+2 i \sqrt {3}}\, x} c_{3} +{\mathrm e}^{-\sqrt {-2+2 i \sqrt {3}}\, x} c_{4} +{\mathrm e}^{-\sqrt {-2-2 i \sqrt {3}}\, x} c_{5} +{\mathrm e}^{\sqrt {-2-2 i \sqrt {3}}\, x} c_{6} \] 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: 53
dsolve(diff(y(x),x$6)-64*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-2 x} \left (\left (c_{4} {\mathrm e}^{3 x}+c_{6} {\mathrm e}^{x}\right ) \cos \left (\sqrt {3}\, x \right )+\left (c_{3} {\mathrm e}^{3 x}+c_{5} {\mathrm e}^{x}\right ) \sin \left (\sqrt {3}\, x \right )+{\mathrm e}^{4 x} c_{1} +c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 68
DSolve[y''''''[x]-64*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-2 x} \left (c_1 e^{4 x}+e^x \left (c_2 e^{2 x}+c_3\right ) \cos \left (\sqrt {3} x\right )+e^x \left (c_6 e^{2 x}+c_5\right ) \sin \left (\sqrt {3} x\right )+c_4\right ) \]