Internal problem ID [1471]
Internal file name [OUTPUT/1472_Sunday_June_05_2022_02_19_04_AM_89291697/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 9 Introduction to Linear Higher Order Equations. Section 9.2. constant
coefficient. Page 483
Problem number: section 9.2, problem 7.
ODE order: 3.
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
[[_3rd_order, _missing_x]]
\[ \boxed {27 y^{\prime \prime \prime }+27 y^{\prime \prime }+9 y^{\prime }+y=0} \] The characteristic equation is \[ 27 \lambda ^{3}+27 \lambda ^{2}+9 \lambda +1 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -{\frac {1}{3}}\\ \lambda _2 &= -{\frac {1}{3}}\\ \lambda _3 &= -{\frac {1}{3}} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{-\frac {x}{3}} c_{1} +x \,{\mathrm e}^{-\frac {x}{3}} c_{2} +x^{2} {\mathrm e}^{-\frac {x}{3}} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-\frac {x}{3}}\\ y_2 &= x \,{\mathrm e}^{-\frac {x}{3}}\\ y_3 &= x^{2} {\mathrm e}^{-\frac {x}{3}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-\frac {x}{3}} c_{1} +x \,{\mathrm e}^{-\frac {x}{3}} c_{2} +x^{2} {\mathrm e}^{-\frac {x}{3}} c_{3} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{-\frac {x}{3}} c_{1} +x \,{\mathrm e}^{-\frac {x}{3}} c_{2} +x^{2} {\mathrm e}^{-\frac {x}{3}} c_{3} \] Verified OK.
Maple trace
`Methods for third 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: 19
dsolve(27*diff(y(x),x$3)+27*diff(y(x),x$2)+9*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {x}{3}} \left (c_{3} x^{2}+c_{2} x +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 25
DSolve[27*y'''[x]+27*y''[x]+9*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-x/3} (x (c_3 x+c_2)+c_1) \]