Internal problem ID [4597]
Internal file name [OUTPUT/4090_Sunday_June_05_2022_12_20_59_PM_45343476/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 20.27, 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]]
\[ \boxed {y^{\prime \prime \prime \prime }+4 y^{\prime \prime }+4 y=0} \] The characteristic equation is \[ \lambda ^{4}+4 \lambda ^{2}+4 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= i \sqrt {2}\\ \lambda _2 &= -i \sqrt {2}\\ \lambda _3 &= i \sqrt {2}\\ \lambda _4 &= -i \sqrt {2} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{i \sqrt {2}\, x} c_{1} +x \,{\mathrm e}^{i \sqrt {2}\, x} c_{2} +{\mathrm e}^{-i \sqrt {2}\, x} c_{3} +x \,{\mathrm e}^{-i \sqrt {2}\, x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{i \sqrt {2}\, x}\\ y_2 &= x \,{\mathrm e}^{i \sqrt {2}\, x}\\ y_3 &= {\mathrm e}^{-i \sqrt {2}\, x}\\ y_4 &= x \,{\mathrm e}^{-i \sqrt {2}\, x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{i \sqrt {2}\, x} c_{1} +x \,{\mathrm e}^{i \sqrt {2}\, x} c_{2} +{\mathrm e}^{-i \sqrt {2}\, x} c_{3} +x \,{\mathrm e}^{-i \sqrt {2}\, x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{i \sqrt {2}\, x} c_{1} +x \,{\mathrm e}^{i \sqrt {2}\, x} c_{2} +{\mathrm e}^{-i \sqrt {2}\, x} c_{3} +x \,{\mathrm e}^{-i \sqrt {2}\, x} c_{4} \] 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: 29
dsolve(diff(y(x),x$4)+4*diff(y(x),x$2)+4*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (x c_{4} +c_{2} \right ) \cos \left (x \sqrt {2}\right )+\sin \left (x \sqrt {2}\right ) \left (c_{3} x +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 38
DSolve[y''''[x]+4*y''[x]+4*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (c_2 x+c_1) \cos \left (\sqrt {2} x\right )+(c_4 x+c_3) \sin \left (\sqrt {2} x\right ) \]