Internal problem ID [5886]
Internal file name [OUTPUT/5134_Sunday_June_05_2022_03_25_49_PM_79468662/index.tex]
Book: THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS.
K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section: Chapter 3. Ordinary Differential Equations. Section 3.6 Summary and Problems. Page
218
Problem number: Problem 3.19.
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]]
\[ \boxed {a^{2} y^{\prime \prime \prime \prime }-y^{\prime \prime }=0} \] The characteristic equation is \[ a^{2} \lambda ^{4}-\lambda ^{2} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 0\\ \lambda _3 &= \frac {1}{a}\\ \lambda _4 &= -\frac {1}{a} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{2} x +c_{1} +{\mathrm e}^{-\frac {x}{a}} c_{3} +{\mathrm e}^{\frac {x}{a}} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= 1\\ y_2 &= x\\ y_3 &= {\mathrm e}^{-\frac {x}{a}}\\ y_4 &= {\mathrm e}^{\frac {x}{a}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} x +c_{1} +{\mathrm e}^{-\frac {x}{a}} c_{3} +{\mathrm e}^{\frac {x}{a}} c_{4} \\ \end{align*}
Verification of solutions
\[ y = c_{2} x +c_{1} +{\mathrm e}^{-\frac {x}{a}} c_{3} +{\mathrm e}^{\frac {x}{a}} c_{4} \] 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: 0.016 (sec). Leaf size: 26
dsolve(a^2*diff(y(x),x$4)=diff(y(x),x$2),y(x), singsol=all)
\[ y \left (x \right ) = c_{1} +c_{2} x +c_{3} {\mathrm e}^{\frac {x}{a}}+c_{4} {\mathrm e}^{-\frac {x}{a}} \]
✓ Solution by Mathematica
Time used: 0.071 (sec). Leaf size: 38
DSolve[a^2*y''''[x]==y''[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to a^2 e^{-\frac {x}{a}} \left (c_1 e^{\frac {2 x}{a}}+c_2\right )+c_4 x+c_3 \]