Internal problem ID [14642]
Internal file name [OUTPUT/14322_Wednesday_April_03_2024_02_17_18_PM_34729963/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: 24.
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]]
\[ \boxed {6 y^{\prime \prime \prime }-5 y^{\prime \prime }-2 y^{\prime }+y=0} \] The characteristic equation is \[ 6 \lambda ^{3}-5 \lambda ^{2}-2 \lambda +1 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 1\\ \lambda _2 &= {\frac {1}{3}}\\ \lambda _3 &= -{\frac {1}{2}} \end {align*}
Therefore the homogeneous solution is \[ y_h(t)=c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{\frac {t}{3}}+{\mathrm e}^{-\frac {t}{2}} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{t}\\ y_2 &= {\mathrm e}^{\frac {t}{3}}\\ y_3 &= {\mathrm e}^{-\frac {t}{2}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{\frac {t}{3}}+{\mathrm e}^{-\frac {t}{2}} c_{3} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{\frac {t}{3}}+{\mathrm e}^{-\frac {t}{2}} c_{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 6 y^{\prime \prime \prime }-5 y^{\prime \prime }-2 y^{\prime }+y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & y^{\prime \prime \prime } \\ \bullet & {} & \textrm {Isolate 3rd derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }=\frac {5 y^{\prime \prime }}{6}+\frac {y^{\prime }}{3}-\frac {y}{6} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }-\frac {5 y^{\prime \prime }}{6}-\frac {y^{\prime }}{3}+\frac {y}{6}=0 \\ \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 )=\frac {5 y_{3}\left (t \right )}{6}+\frac {y_{2}\left (t \right )}{3}-\frac {y_{1}\left (t \right )}{6} \\ & {} & \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 )=\frac {5 y_{3}\left (t \right )}{6}+\frac {y_{2}\left (t \right )}{3}-\frac {y_{1}\left (t \right )}{6}\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 \\ -\frac {1}{6} & \frac {1}{3} & \frac {5}{6} \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 \\ -\frac {1}{6} & \frac {1}{3} & \frac {5}{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 [-\frac {1}{2}, \left [\begin {array}{c} 4 \\ -2 \\ 1 \end {array}\right ]\right ], \left [\frac {1}{3}, \left [\begin {array}{c} 9 \\ 3 \\ 1 \end {array}\right ]\right ], \left [1, \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {1}{2}, \left [\begin {array}{c} 4 \\ -2 \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{-\frac {t}{2}}\cdot \left [\begin {array}{c} 4 \\ -2 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [\frac {1}{3}, \left [\begin {array}{c} 9 \\ 3 \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{2}={\mathrm e}^{\frac {t}{3}}\cdot \left [\begin {array}{c} 9 \\ 3 \\ 1 \end {array}\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}}_{3}={\mathrm e}^{t}\cdot \left [\begin {array}{c} 1 \\ 1 \\ 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} \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}=c_{1} {\mathrm e}^{-\frac {t}{2}}\cdot \left [\begin {array}{c} 4 \\ -2 \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{\frac {t}{3}}\cdot \left [\begin {array}{c} 9 \\ 3 \\ 1 \end {array}\right ]+c_{3} {\mathrm e}^{t}\cdot \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\left (c_{3} {\mathrm e}^{\frac {3 t}{2}}+9 c_{2} {\mathrm e}^{\frac {5 t}{6}}+4 c_{1} \right ) {\mathrm e}^{-\frac {t}{2}} \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: 23
dsolve(6*diff(y(t),t$3)-5*diff(y(t),t$2)-2*diff(y(t),t)+y(t)=0,y(t), singsol=all)
\[ y \left (t \right ) = \left (c_{3} {\mathrm e}^{\frac {3 t}{2}}+c_{2} {\mathrm e}^{\frac {5 t}{6}}+c_{1} \right ) {\mathrm e}^{-\frac {t}{2}} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 32
DSolve[6*y'''[t]-5*y''[t]-2*y'[t]+y[t]==0,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to c_1 e^{t/3}+c_2 e^{-t/2}+c_3 e^t \]