Internal problem ID [4580]
Internal file name [OUTPUT/4073_Sunday_June_05_2022_12_18_51_PM_53894104/index.tex]
Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 4. Higher order linear differential equations. Lesson 20. Constant
coefficients
Problem number: Exercise 20.10, page 220.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }-y a^{2}=0} \] The characteristic equation is \[ \lambda ^{4}-a^{2} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= \sqrt {a}\\ \lambda _2 &= -\sqrt {a}\\ \lambda _3 &= \sqrt {-a}\\ \lambda _4 &= -\sqrt {-a} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{-\sqrt {-a}\, x} c_{1} +{\mathrm e}^{-\sqrt {a}\, x} c_{2} +{\mathrm e}^{\sqrt {-a}\, x} c_{3} +{\mathrm e}^{\sqrt {a}\, x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-\sqrt {-a}\, x}\\ y_2 &= {\mathrm e}^{-\sqrt {a}\, x}\\ y_3 &= {\mathrm e}^{\sqrt {-a}\, x}\\ y_4 &= {\mathrm e}^{\sqrt {a}\, x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-\sqrt {-a}\, x} c_{1} +{\mathrm e}^{-\sqrt {a}\, x} c_{2} +{\mathrm e}^{\sqrt {-a}\, x} c_{3} +{\mathrm e}^{\sqrt {a}\, x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{-\sqrt {-a}\, x} c_{1} +{\mathrm e}^{-\sqrt {a}\, x} c_{2} +{\mathrm e}^{\sqrt {-a}\, x} c_{3} +{\mathrm e}^{\sqrt {a}\, x} c_{4} \] 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: 0.0 (sec). Leaf size: 38
dsolve(diff(y(x),x$4)-a^2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{\sqrt {a}\, x}+c_{2} {\mathrm e}^{-\sqrt {a}\, x}+c_{3} \sin \left (\sqrt {a}\, x \right )+c_{4} \cos \left (\sqrt {a}\, x \right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 53
DSolve[y''''[x]-a^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 e^{-\sqrt {a} x}+c_4 e^{\sqrt {a} x}+c_1 \cos \left (\sqrt {a} x\right )+c_3 \sin \left (\sqrt {a} x\right ) \]