Internal problem ID [4586]
Internal file name [OUTPUT/4079_Sunday_June_05_2022_12_19_34_PM_87531353/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 20.16, page 220.
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]]
\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime }-8 y=0} \] The characteristic equation is \[ \lambda ^{3}-6 \lambda ^{2}+12 \lambda -8 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 2\\ \lambda _2 &= 2\\ \lambda _3 &= 2 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x} c_{2} +x^{2} {\mathrm e}^{2 x} c_{3} \] 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} x\\ y_3 &= 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}+x \,{\mathrm e}^{2 x} c_{2} +x^{2} {\mathrm e}^{2 x} c_{3} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x} c_{2} +x^{2} {\mathrm e}^{2 x} c_{3} \] 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: 0.0 (sec). Leaf size: 19
dsolve(diff(y(x),x$3)-6*diff(y(x),x$2)+12*diff(y(x),x)-8*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{2 x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 23
DSolve[y'''[x]-6*y''[x]+12*y'[x]-8*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{2 x} (x (c_3 x+c_2)+c_1) \]