Internal problem ID [14669]
Internal file name [OUTPUT/14349_Wednesday_April_03_2024_02_17_26_PM_98292656/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 4. Higher Order Equations. Exercises 4.5, page 175
Problem number: 58.
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]]
\[ \boxed {y^{\prime \prime \prime }+9 y^{\prime \prime }+16 y^{\prime }-26 y=0} \] The characteristic equation is \[ \lambda ^{3}+9 \lambda ^{2}+16 \lambda -26 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 1\\ \lambda _2 &= -5-i\\ \lambda _3 &= -5+i \end {align*}
Therefore the homogeneous solution is \[ y_h(t)=c_{1} {\mathrm e}^{t}+{\mathrm e}^{\left (-5-i\right ) t} c_{2} +{\mathrm e}^{\left (-5+i\right ) t} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{t}\\ y_2 &= {\mathrm e}^{\left (-5-i\right ) t}\\ y_3 &= {\mathrm e}^{\left (-5+i\right ) t} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{t}+{\mathrm e}^{\left (-5-i\right ) t} c_{2} +{\mathrm e}^{\left (-5+i\right ) t} c_{3} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{t}+{\mathrm e}^{\left (-5-i\right ) t} c_{2} +{\mathrm e}^{\left (-5+i\right ) t} c_{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }+9 y^{\prime \prime }+16 y^{\prime }-26 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & y^{\prime \prime \prime } \\ \square & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{1}\left (t \right ) \\ {} & {} & y_{1}\left (t \right )=y \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{2}\left (t \right ) \\ {} & {} & y_{2}\left (t \right )=y^{\prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{3}\left (t \right ) \\ {} & {} & y_{3}\left (t \right )=y^{\prime \prime } \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} y_{3}^{\prime }\left (t \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{3}^{\prime }\left (t \right )=-9 y_{3}\left (t \right )-16 y_{2}\left (t \right )+26 y_{1}\left (t \right ) \\ & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & {} & \left [y_{2}\left (t \right )=y_{1}^{\prime }\left (t \right ), y_{3}\left (t \right )=y_{2}^{\prime }\left (t \right ), y_{3}^{\prime }\left (t \right )=-9 y_{3}\left (t \right )-16 y_{2}\left (t \right )+26 y_{1}\left (t \right )\right ] \\ \bullet & {} & \textrm {Define vector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}\left (t \right )=\left [\begin {array}{c} y_{1}\left (t \right ) \\ y_{2}\left (t \right ) \\ y_{3}\left (t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (t \right )=\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 26 & -16 & -9 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (t \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 26 & -16 & -9 \end {array}\right ] \\ \bullet & {} & \textrm {Rewrite the system as}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (t \right )=A \cdot {\moverset {\rightarrow }{y}}\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 [1, \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ]\right ], \left [-5-\mathrm {I}, \left [\begin {array}{c} \frac {6}{169}-\frac {5 \,\mathrm {I}}{338} \\ -\frac {5}{26}+\frac {\mathrm {I}}{26} \\ 1 \end {array}\right ]\right ], \left [-5+\mathrm {I}, \left [\begin {array}{c} \frac {6}{169}+\frac {5 \,\mathrm {I}}{338} \\ -\frac {5}{26}-\frac {\mathrm {I}}{26} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [1, \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{t}\cdot \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-5-\mathrm {I}, \left [\begin {array}{c} \frac {6}{169}-\frac {5 \,\mathrm {I}}{338} \\ -\frac {5}{26}+\frac {\mathrm {I}}{26} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-5-\mathrm {I}\right ) t}\cdot \left [\begin {array}{c} \frac {6}{169}-\frac {5 \,\mathrm {I}}{338} \\ -\frac {5}{26}+\frac {\mathrm {I}}{26} \\ 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}^{-5 t}\cdot \left (\cos \left (t \right )-\mathrm {I} \sin \left (t \right )\right )\cdot \left [\begin {array}{c} \frac {6}{169}-\frac {5 \,\mathrm {I}}{338} \\ -\frac {5}{26}+\frac {\mathrm {I}}{26} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-5 t}\cdot \left [\begin {array}{c} \left (\frac {6}{169}-\frac {5 \,\mathrm {I}}{338}\right ) \left (\cos \left (t \right )-\mathrm {I} \sin \left (t \right )\right ) \\ \left (-\frac {5}{26}+\frac {\mathrm {I}}{26}\right ) \left (\cos \left (t \right )-\mathrm {I} \sin \left (t \right )\right ) \\ \cos \left (t \right )-\mathrm {I} \sin \left (t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{2}\left (t \right )={\mathrm e}^{-5 t}\cdot \left [\begin {array}{c} \frac {6 \cos \left (t \right )}{169}-\frac {5 \sin \left (t \right )}{338} \\ -\frac {5 \cos \left (t \right )}{26}+\frac {\sin \left (t \right )}{26} \\ \cos \left (t \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{3}\left (t \right )={\mathrm e}^{-5 t}\cdot \left [\begin {array}{c} -\frac {6 \sin \left (t \right )}{169}-\frac {5 \cos \left (t \right )}{338} \\ \frac {5 \sin \left (t \right )}{26}+\frac {\cos \left (t \right )}{26} \\ -\sin \left (t \right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {General solution to the system of ODEs}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}=c_{1} {\moverset {\rightarrow }{y}}_{1}+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (t \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (t \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}=c_{1} {\mathrm e}^{t}\cdot \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{-5 t}\cdot \left [\begin {array}{c} \frac {6 \cos \left (t \right )}{169}-\frac {5 \sin \left (t \right )}{338} \\ -\frac {5 \cos \left (t \right )}{26}+\frac {\sin \left (t \right )}{26} \\ \cos \left (t \right ) \end {array}\right ]+c_{3} {\mathrm e}^{-5 t}\cdot \left [\begin {array}{c} -\frac {6 \sin \left (t \right )}{169}-\frac {5 \cos \left (t \right )}{338} \\ \frac {5 \sin \left (t \right )}{26}+\frac {\cos \left (t \right )}{26} \\ -\sin \left (t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {\left (338 c_{1} {\mathrm e}^{6 t}+\left (12 c_{2} -5 c_{3} \right ) \cos \left (t \right )-5 \sin \left (t \right ) \left (c_{2} +\frac {12 c_{3}}{5}\right )\right ) {\mathrm e}^{-5 t}}{338} \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: 24
dsolve(diff(y(t),t$3)+9*diff(y(t),t$2)+16*diff(y(t),t)-26*y(t)=0,y(t), singsol=all)
\[ y \left (t \right ) = \left ({\mathrm e}^{6 t} c_{1} +\sin \left (t \right ) c_{2} +\cos \left (t \right ) c_{3} \right ) {\mathrm e}^{-5 t} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 30
DSolve[y'''[t]+9*y''[t]+16*y'[t]-26*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to e^{-5 t} \left (c_3 e^{6 t}+c_2 \cos (t)+c_1 \sin (t)\right ) \]