Internal problem ID [6356]
Internal file name [OUTPUT/5604_Sunday_June_05_2022_03_44_38_PM_42445937/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 2. Section 2.7. HIGHER ORDER LINEAR EQUATIONS, COUPLED
HARMONIC OSCILLATORS Page 98
Problem number: 9.
ODE order: 4.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }-2 a^{2} y^{\prime \prime }+a^{4} y=0} \] The characteristic equation is \[ a^{4}-2 a^{2} \lambda ^{2}+\lambda ^{4} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= a\\ \lambda _2 &= a\\ \lambda _3 &= -a\\ \lambda _4 &= -a \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{-a x}+x \,{\mathrm e}^{-a x} c_{2} +{\mathrm e}^{a x} c_{3} +x \,{\mathrm e}^{a x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-a x}\\ y_2 &= x \,{\mathrm e}^{-a x}\\ y_3 &= {\mathrm e}^{a x}\\ y_4 &= x \,{\mathrm e}^{a x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{-a x}+x \,{\mathrm e}^{-a x} c_{2} +{\mathrm e}^{a x} c_{3} +x \,{\mathrm e}^{a x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{-a x}+x \,{\mathrm e}^{-a x} c_{2} +{\mathrm e}^{a x} c_{3} +x \,{\mathrm e}^{a x} c_{4} \] 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: 0.015 (sec). Leaf size: 26
dsolve(diff(y(x),x$4)-2*a^2*diff(y(x),x$2)+a^4*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{4} x +c_{3} \right ) {\mathrm e}^{-a x}+{\mathrm e}^{a x} \left (c_{2} x +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 38
DSolve[y''''[x]-2*a^2*y''[x]+a^4*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-a x} \left (c_3 e^{2 a x}+x \left (c_4 e^{2 a x}+c_2\right )+c_1\right ) \]