Internal problem ID [14648]
Internal file name [OUTPUT/14328_Wednesday_April_03_2024_02_17_20_PM_35630610/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: 30.
ODE order: 4.
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
[[_high_order, _missing_x]]
\[ \boxed {y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }-y^{\prime \prime }+54 y^{\prime }-72 y=0} \] The characteristic equation is \[ \lambda ^{4}-6 \lambda ^{3}-\lambda ^{2}+54 \lambda -72 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 2\\ \lambda _2 &= 3\\ \lambda _3 &= 4\\ \lambda _4 &= -3 \end {align*}
Therefore the homogeneous solution is \[ y_h(t)=c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{3 t}+{\mathrm e}^{4 t} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-3 t}\\ y_2 &= {\mathrm e}^{2 t}\\ y_3 &= {\mathrm e}^{3 t}\\ y_4 &= {\mathrm e}^{4 t} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{3 t}+{\mathrm e}^{4 t} c_{4} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{3 t}+{\mathrm e}^{4 t} c_{4} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }-y^{\prime \prime }+54 y^{\prime }-72 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & y^{\prime \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 {Define new variable}\hspace {3pt} y_{4}\left (t \right ) \\ {} & {} & y_{4}\left (t \right )=y^{\prime \prime \prime } \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} y_{4}^{\prime }\left (t \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{4}^{\prime }\left (t \right )=6 y_{4}\left (t \right )+y_{3}\left (t \right )-54 y_{2}\left (t \right )+72 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_{4}\left (t \right )=y_{3}^{\prime }\left (t \right ), y_{4}^{\prime }\left (t \right )=6 y_{4}\left (t \right )+y_{3}\left (t \right )-54 y_{2}\left (t \right )+72 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 ) \\ y_{4}\left (t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (t \right )=\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 72 & -54 & 1 & 6 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (t \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 72 & -54 & 1 & 6 \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 [-3, \left [\begin {array}{c} -\frac {1}{27} \\ \frac {1}{9} \\ -\frac {1}{3} \\ 1 \end {array}\right ]\right ], \left [2, \left [\begin {array}{c} \frac {1}{8} \\ \frac {1}{4} \\ \frac {1}{2} \\ 1 \end {array}\right ]\right ], \left [3, \left [\begin {array}{c} \frac {1}{27} \\ \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]\right ], \left [4, \left [\begin {array}{c} \frac {1}{64} \\ \frac {1}{16} \\ \frac {1}{4} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-3, \left [\begin {array}{c} -\frac {1}{27} \\ \frac {1}{9} \\ -\frac {1}{3} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{-3 t}\cdot \left [\begin {array}{c} -\frac {1}{27} \\ \frac {1}{9} \\ -\frac {1}{3} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [2, \left [\begin {array}{c} \frac {1}{8} \\ \frac {1}{4} \\ \frac {1}{2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{2}={\mathrm e}^{2 t}\cdot \left [\begin {array}{c} \frac {1}{8} \\ \frac {1}{4} \\ \frac {1}{2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [3, \left [\begin {array}{c} \frac {1}{27} \\ \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{3}={\mathrm e}^{3 t}\cdot \left [\begin {array}{c} \frac {1}{27} \\ \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [4, \left [\begin {array}{c} \frac {1}{64} \\ \frac {1}{16} \\ \frac {1}{4} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{4}={\mathrm e}^{4 t}\cdot \left [\begin {array}{c} \frac {1}{64} \\ \frac {1}{16} \\ \frac {1}{4} \\ 1 \end {array}\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}+c_{3} {\moverset {\rightarrow }{y}}_{3}+c_{4} {\moverset {\rightarrow }{y}}_{4} \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}=c_{1} {\mathrm e}^{-3 t}\cdot \left [\begin {array}{c} -\frac {1}{27} \\ \frac {1}{9} \\ -\frac {1}{3} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{2 t}\cdot \left [\begin {array}{c} \frac {1}{8} \\ \frac {1}{4} \\ \frac {1}{2} \\ 1 \end {array}\right ]+c_{3} {\mathrm e}^{3 t}\cdot \left [\begin {array}{c} \frac {1}{27} \\ \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]+{\mathrm e}^{4 t} c_{4} \cdot \left [\begin {array}{c} \frac {1}{64} \\ \frac {1}{16} \\ \frac {1}{4} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=-\frac {{\mathrm e}^{-3 t} \left (-\frac {27 c_{2} {\mathrm e}^{5 t}}{8}-c_{3} {\mathrm e}^{6 t}-\frac {27 \,{\mathrm e}^{7 t} c_{4}}{64}+c_{1} \right )}{27} \end {array} \]
Maple trace
`Methods for high 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: 29
dsolve(diff(y(t),t$4)-6*diff(y(t),t$3)-diff(y(t),t$2)+54*diff(y(t),t)-72*y(t)=0,y(t), singsol=all)
\[ y \left (t \right ) = \left ({\mathrm e}^{7 t} c_{2} +c_{3} {\mathrm e}^{6 t}+c_{4} {\mathrm e}^{5 t}+c_{1} \right ) {\mathrm e}^{-3 t} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 36
DSolve[y''''[t]-6*y'''[t]-y''[t]+54*y'[t]-72*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to e^{-3 t} \left (e^{5 t} \left (e^t \left (c_4 e^t+c_3\right )+c_2\right )+c_1\right ) \]