Internal problem ID [13555]
Internal file name [OUTPUT/12727_Saturday_February_17_2024_08_44_13_AM_3851012/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 14. Higher order equations and the reduction of order method. Additional
exercises page 277
Problem number: 14.5 (a).
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]]
\[ \boxed {y^{\prime \prime \prime }-9 y^{\prime \prime }+27 y^{\prime }-27 y=0} \] The characteristic equation is \[ \lambda ^{3}-9 \lambda ^{2}+27 \lambda -27 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 3\\ \lambda _2 &= 3\\ \lambda _3 &= 3 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{3 x}+x \,{\mathrm e}^{3 x} c_{2} +x^{2} {\mathrm e}^{3 x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{3 x}\\ y_2 &= {\mathrm e}^{3 x} x\\ y_3 &= x^{2} {\mathrm e}^{3 x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{3 x}+x \,{\mathrm e}^{3 x} c_{2} +x^{2} {\mathrm e}^{3 x} c_{3} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{3 x}+x \,{\mathrm e}^{3 x} c_{2} +x^{2} {\mathrm e}^{3 x} c_{3} \] 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)-9*diff(y(x),x$2)+27*diff(y(x),x)-27*y(x)=0,exp(3*x)],singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{3 x} \left (x^{2} c_{3} +c_{2} x +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 23
DSolve[y'''[x]-9*y''[x]+27*y'[x]-27*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{3 x} (x (c_3 x+c_2)+c_1) \]