Internal problem ID [6353]
Internal file name [OUTPUT/5601_Sunday_June_05_2022_03_44_35_PM_64782230/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: 6.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+6 y^{\prime \prime }+4 y^{\prime }+y=0} \] The characteristic equation is \[ \lambda ^{4}+4 \lambda ^{3}+6 \lambda ^{2}+4 \lambda +1 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -1\\ \lambda _2 &= -1\\ \lambda _3 &= -1\\ \lambda _4 &= -1 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{-x}+c_{2} x \,{\mathrm e}^{-x}+c_{3} {\mathrm e}^{-x} x^{2}+x^{3} {\mathrm e}^{-x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-x}\\ y_2 &= x \,{\mathrm e}^{-x}\\ y_3 &= x^{2} {\mathrm e}^{-x}\\ y_4 &= x^{3} {\mathrm e}^{-x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{-x}+c_{2} x \,{\mathrm e}^{-x}+c_{3} {\mathrm e}^{-x} x^{2}+x^{3} {\mathrm e}^{-x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{-x}+c_{2} x \,{\mathrm e}^{-x}+c_{3} {\mathrm e}^{-x} x^{2}+x^{3} {\mathrm e}^{-x} c_{4} \] 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: 24
dsolve(diff(y(x),x$4)+4*diff(y(x),x$3)+6*diff(y(x),x$2)+4*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-x} \left (c_{4} x^{3}+c_{3} x^{2}+c_{2} x +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 28
DSolve[y''''[x]+4*y'''[x]+6*y''[x]+4*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-x} (x (x (c_4 x+c_3)+c_2)+c_1) \]