Internal problem ID [13640]
Internal file name [OUTPUT/12812_Saturday_February_17_2024_08_44_56_AM_47089625/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 19. Arbitrary Homogeneous linear equations with constant coefficients.
Additional exercises page 369
Problem number: 19.4 (i).
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]]
\[ \boxed {y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+16 y^{\prime }-16 y=0} \] The characteristic equation is \[ \lambda ^{4}-4 \lambda ^{3}+16 \lambda -16 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -2\\ \lambda _2 &= 2\\ \lambda _3 &= 2\\ \lambda _4 &= 2 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x} c_{3} +x^{2} {\mathrm e}^{2 x} c_{4} \] 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 &= x \,{\mathrm e}^{2 x}\\ y_4 &= x^{2} {\mathrm e}^{2 x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x} c_{3} +x^{2} {\mathrm e}^{2 x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x} c_{3} +x^{2} {\mathrm e}^{2 x} c_{4} \] 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: 26
dsolve(diff(y(x),x$4)-4*diff(y(x),x$3)+16*diff(y(x),x)-16*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{4} x^{2}+c_{3} x +c_{2} \right ) {\mathrm e}^{2 x}+{\mathrm e}^{-2 x} c_{1} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 32
DSolve[y''''[x]-4*y'''[x]+16*y'[x]-16*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-2 x} \left (e^{4 x} (x (c_4 x+c_3)+c_2)+c_1\right ) \]