14.6 problem 1(f)

Internal problem ID [11505]
Internal file name [OUTPUT/10488_Thursday_May_18_2023_04_20_54_AM_14995609/index.tex]

Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section: Chapter 2, Second order linear equations. Section 2.5 Higher order equations. Exercises page 130
Problem number: 1(f).
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]]

\[ \boxed {x^{\prime \prime \prime }-8 x=0} \] The characteristic equation is \[ \lambda ^{3}-8 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 2\\ \lambda _2 &= i \sqrt {3}-1\\ \lambda _3 &= -i \sqrt {3}-1 \end {align*}

Therefore the homogeneous solution is \[ x_h(t)={\mathrm e}^{2 t} c_{1} +{\mathrm e}^{\left (-i \sqrt {3}-1\right ) t} c_{2} +{\mathrm e}^{\left (i \sqrt {3}-1\right ) t} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} x_1 &= {\mathrm e}^{2 t}\\ x_2 &= {\mathrm e}^{\left (-i \sqrt {3}-1\right ) t}\\ x_3 &= {\mathrm e}^{\left (i \sqrt {3}-1\right ) t} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} x &= {\mathrm e}^{2 t} c_{1} +{\mathrm e}^{\left (-i \sqrt {3}-1\right ) t} c_{2} +{\mathrm e}^{\left (i \sqrt {3}-1\right ) t} c_{3} \\ \end{align*}

Verification of solutions

\[ x = {\mathrm e}^{2 t} c_{1} +{\mathrm e}^{\left (-i \sqrt {3}-1\right ) t} c_{2} +{\mathrm e}^{\left (i \sqrt {3}-1\right ) t} c_{3} \] Verified OK.

14.6.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime \prime \prime }-8 x=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & x^{\prime \prime \prime } \\ \square & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} x_{1}\left (t \right ) \\ {} & {} & x_{1}\left (t \right )=x \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} x_{2}\left (t \right ) \\ {} & {} & x_{2}\left (t \right )=x^{\prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} x_{3}\left (t \right ) \\ {} & {} & x_{3}\left (t \right )=x^{\prime \prime } \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} x_{3}^{\prime }\left (t \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & x_{3}^{\prime }\left (t \right )=8 x_{1}\left (t \right ) \\ & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & {} & \left [x_{2}\left (t \right )=x_{1}^{\prime }\left (t \right ), x_{3}\left (t \right )=x_{2}^{\prime }\left (t \right ), x_{3}^{\prime }\left (t \right )=8 x_{1}\left (t \right )\right ] \\ \bullet & {} & \textrm {Define vector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}\left (t \right )=\left [\begin {array}{c} x_{1}\left (t \right ) \\ x_{2}\left (t \right ) \\ x_{3}\left (t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}^{\prime }\left (t \right )=\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 8 & 0 & 0 \end {array}\right ]\cdot {\moverset {\rightarrow }{x}}\left (t \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 8 & 0 & 0 \end {array}\right ] \\ \bullet & {} & \textrm {Rewrite the system as}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}^{\prime }\left (t \right )=A \cdot {\moverset {\rightarrow }{x}}\left (t \right ) \\ \bullet & {} & \textrm {To solve the system, find the eigenvalues and eigenvectors of}\hspace {3pt} A \\ \bullet & {} & \textrm {Eigenpairs of}\hspace {3pt} A \\ {} & {} & \left [\left [2, \left [\begin {array}{c} \frac {1}{4} \\ \frac {1}{2} \\ 1 \end {array}\right ]\right ], \left [-\mathrm {I} \sqrt {3}-1, \left [\begin {array}{c} \frac {1}{\left (-\mathrm {I} \sqrt {3}-1\right )^{2}} \\ \frac {1}{-\mathrm {I} \sqrt {3}-1} \\ 1 \end {array}\right ]\right ], \left [\mathrm {I} \sqrt {3}-1, \left [\begin {array}{c} \frac {1}{\left (\mathrm {I} \sqrt {3}-1\right )^{2}} \\ \frac {1}{\mathrm {I} \sqrt {3}-1} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [2, \left [\begin {array}{c} \frac {1}{4} \\ \frac {1}{2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}_{1}={\mathrm e}^{2 t}\cdot \left [\begin {array}{c} \frac {1}{4} \\ \frac {1}{2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-\mathrm {I} \sqrt {3}-1, \left [\begin {array}{c} \frac {1}{\left (-\mathrm {I} \sqrt {3}-1\right )^{2}} \\ \frac {1}{-\mathrm {I} \sqrt {3}-1} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-\mathrm {I} \sqrt {3}-1\right ) t}\cdot \left [\begin {array}{c} \frac {1}{\left (-\mathrm {I} \sqrt {3}-1\right )^{2}} \\ \frac {1}{-\mathrm {I} \sqrt {3}-1} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Use Euler identity to write solution in terms of}\hspace {3pt} \sin \hspace {3pt}\textrm {and}\hspace {3pt} \cos \\ {} & {} & {\mathrm e}^{-t}\cdot \left (\cos \left (\sqrt {3}\, t \right )-\mathrm {I} \sin \left (\sqrt {3}\, t \right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (-\mathrm {I} \sqrt {3}-1\right )^{2}} \\ \frac {1}{-\mathrm {I} \sqrt {3}-1} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-t}\cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {3}\, t \right )-\mathrm {I} \sin \left (\sqrt {3}\, t \right )}{\left (-\mathrm {I} \sqrt {3}-1\right )^{2}} \\ \frac {\cos \left (\sqrt {3}\, t \right )-\mathrm {I} \sin \left (\sqrt {3}\, t \right )}{-\mathrm {I} \sqrt {3}-1} \\ \cos \left (\sqrt {3}\, t \right )-\mathrm {I} \sin \left (\sqrt {3}\, t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{x}}_{2}\left (t \right )={\mathrm e}^{-t}\cdot \left [\begin {array}{c} -\frac {\cos \left (\sqrt {3}\, t \right )}{8}-\frac {\sin \left (\sqrt {3}\, t \right ) \sqrt {3}}{8} \\ -\frac {\cos \left (\sqrt {3}\, t \right )}{4}+\frac {\sin \left (\sqrt {3}\, t \right ) \sqrt {3}}{4} \\ \cos \left (\sqrt {3}\, t \right ) \end {array}\right ], {\moverset {\rightarrow }{x}}_{3}\left (t \right )={\mathrm e}^{-t}\cdot \left [\begin {array}{c} -\frac {\cos \left (\sqrt {3}\, t \right ) \sqrt {3}}{8}+\frac {\sin \left (\sqrt {3}\, t \right )}{8} \\ \frac {\cos \left (\sqrt {3}\, t \right ) \sqrt {3}}{4}+\frac {\sin \left (\sqrt {3}\, t \right )}{4} \\ -\sin \left (\sqrt {3}\, t \right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {General solution to the system of ODEs}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}=c_{1} {\moverset {\rightarrow }{x}}_{1}+c_{2} {\moverset {\rightarrow }{x}}_{2}\left (t \right )+c_{3} {\moverset {\rightarrow }{x}}_{3}\left (t \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}={\mathrm e}^{2 t} c_{1} \cdot \left [\begin {array}{c} \frac {1}{4} \\ \frac {1}{2} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{-t}\cdot \left [\begin {array}{c} -\frac {\cos \left (\sqrt {3}\, t \right )}{8}-\frac {\sin \left (\sqrt {3}\, t \right ) \sqrt {3}}{8} \\ -\frac {\cos \left (\sqrt {3}\, t \right )}{4}+\frac {\sin \left (\sqrt {3}\, t \right ) \sqrt {3}}{4} \\ \cos \left (\sqrt {3}\, t \right ) \end {array}\right ]+c_{3} {\mathrm e}^{-t}\cdot \left [\begin {array}{c} -\frac {\cos \left (\sqrt {3}\, t \right ) \sqrt {3}}{8}+\frac {\sin \left (\sqrt {3}\, t \right )}{8} \\ \frac {\cos \left (\sqrt {3}\, t \right ) \sqrt {3}}{4}+\frac {\sin \left (\sqrt {3}\, t \right )}{4} \\ -\sin \left (\sqrt {3}\, t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & x=-\frac {{\mathrm e}^{-t} \left (c_{3} \sqrt {3}+c_{2} \right ) \cos \left (\sqrt {3}\, t \right )}{8}-\frac {{\mathrm e}^{-t} \left (c_{2} \sqrt {3}-c_{3} \right ) \sin \left (\sqrt {3}\, t \right )}{8}+\frac {{\mathrm e}^{2 t} c_{1}}{4} \end {array} \]

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: 35

dsolve(diff(x(t),t$3)-8*x(t)=0,x(t), singsol=all)
 

\[ x \left (t \right ) = c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{-t} \sin \left (\sqrt {3}\, t \right )+c_{3} {\mathrm e}^{-t} \cos \left (\sqrt {3}\, t \right ) \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 52

DSolve[x'''[t]-x[t]==0,x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to e^{-t/2} \left (c_1 e^{3 t/2}+c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )+c_3 \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \]