14.22 problem 22

Internal problem ID [14696]
Internal file name [OUTPUT/14376_Wednesday_April_03_2024_02_17_49_PM_81451068/index.tex]

Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section: Chapter 4. Higher Order Equations. Exercises 4.6, page 187
Problem number: 22.
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, _linear, _nonhomogeneous]]

\[ \boxed {y^{\prime \prime \prime }-13 y^{\prime }+12 y=\cos \left (t \right )} \] This is higher order nonhomogeneous ODE. Let the solution be \[ y = y_h + y_p \] Where \(y_h\) is the solution to the homogeneous ODE And \(y_p\) is a particular solution to the nonhomogeneous ODE. \(y_h\) is the solution to \[ y^{\prime \prime \prime }-13 y^{\prime }+12 y = 0 \] The characteristic equation is \[ \lambda ^{3}-13 \lambda +12 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 1\\ \lambda _2 &= 3\\ \lambda _3 &= -4 \end {align*}

Therefore the homogeneous solution is \[ y_h(t)=c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-4 t}+c_{3} {\mathrm e}^{3 t} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{t} \\ y_2 &= {\mathrm e}^{-4 t} \\ y_3 &= {\mathrm e}^{3 t} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }-13 y^{\prime }+12 y = \cos \left (t \right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ \cos \left (t \right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{\cos \left (t \right ), \sin \left (t \right )\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{{\mathrm e}^{t}, {\mathrm e}^{-4 t}, {\mathrm e}^{3 t}\} \] Since there is no duplication between the basis function in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis in the UC_set. \[ y_p = A_{1} \cos \left (t \right )+A_{2} \sin \left (t \right ) \] The unknowns \(\{A_{1}, A_{2}\}\) are found by substituting the above trial solution \(y_p\) into the ODE and comparing coefficients. Substituting the trial solution into the ODE and simplifying gives \[ 14 A_{1} \sin \left (t \right )-14 A_{2} \cos \left (t \right )+12 A_{1} \cos \left (t \right )+12 A_{2} \sin \left (t \right ) = \cos \left (t \right ) \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = {\frac {3}{85}}, A_{2} = -{\frac {7}{170}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {3 \cos \left (t \right )}{85}-\frac {7 \sin \left (t \right )}{170} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-4 t}+c_{3} {\mathrm e}^{3 t}\right ) + \left (\frac {3 \cos \left (t \right )}{85}-\frac {7 \sin \left (t \right )}{170}\right ) \\ \end{align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-4 t}+c_{3} {\mathrm e}^{3 t}+\frac {3 \cos \left (t \right )}{85}-\frac {7 \sin \left (t \right )}{170} \\ \end{align*}

Verification of solutions

\[ y = c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-4 t}+c_{3} {\mathrm e}^{3 t}+\frac {3 \cos \left (t \right )}{85}-\frac {7 \sin \left (t \right )}{170} \] Verified OK.

14.22.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }-13 y^{\prime }+12 y=\cos \left (t \right ) \\ \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 )=\cos \left (t \right )+13 y_{2}\left (t \right )-12 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 )=\cos \left (t \right )+13 y_{2}\left (t \right )-12 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 \\ -12 & 13 & 0 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (t \right )+\left [\begin {array}{c} 0 \\ 0 \\ \cos \left (t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Define the forcing function}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{f}}\left (t \right )=\left [\begin {array}{c} 0 \\ 0 \\ \cos \left (t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -12 & 13 & 0 \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 )+{\moverset {\rightarrow }{f}} \\ \bullet & {} & \textrm {To solve the system, find the eigenvalues and eigenvectors of}\hspace {3pt} A \\ \bullet & {} & \textrm {Eigenpairs of}\hspace {3pt} A \\ {} & {} & \left [\left [-4, \left [\begin {array}{c} \frac {1}{16} \\ -\frac {1}{4} \\ 1 \end {array}\right ]\right ], \left [1, \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ]\right ], \left [3, \left [\begin {array}{c} \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-4, \left [\begin {array}{c} \frac {1}{16} \\ -\frac {1}{4} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{-4 t}\cdot \left [\begin {array}{c} \frac {1}{16} \\ -\frac {1}{4} \\ 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}}_{2}={\mathrm e}^{t}\cdot \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [3, \left [\begin {array}{c} \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}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {General solution of the system of ODEs can be written in terms of the particular solution}\hspace {3pt} {\moverset {\rightarrow }{y}}_{p}\left (t \right ) \\ {} & {} & {\moverset {\rightarrow }{y}}\left (t \right )=c_{1} {\moverset {\rightarrow }{y}}_{1}+c_{2} {\moverset {\rightarrow }{y}}_{2}+c_{3} {\moverset {\rightarrow }{y}}_{3}+{\moverset {\rightarrow }{y}}_{p}\left (t \right ) \\ \square & {} & \textrm {Fundamental matrix}\hspace {3pt} \\ {} & \circ & \textrm {Let}\hspace {3pt} \phi \left (t \right )\hspace {3pt}\textrm {be the matrix whose columns are the independent solutions of the homogeneous system.}\hspace {3pt} \\ {} & {} & \phi \left (t \right )=\left [\begin {array}{ccc} \frac {{\mathrm e}^{-4 t}}{16} & {\mathrm e}^{t} & \frac {{\mathrm e}^{3 t}}{9} \\ -\frac {{\mathrm e}^{-4 t}}{4} & {\mathrm e}^{t} & \frac {{\mathrm e}^{3 t}}{3} \\ {\mathrm e}^{-4 t} & {\mathrm e}^{t} & {\mathrm e}^{3 t} \end {array}\right ] \\ {} & \circ & \textrm {The fundamental matrix,}\hspace {3pt} \Phi \left (t \right )\hspace {3pt}\textrm {is a normalized version of}\hspace {3pt} \phi \left (t \right )\hspace {3pt}\textrm {satisfying}\hspace {3pt} \Phi \left (0\right )=I \hspace {3pt}\textrm {where}\hspace {3pt} I \hspace {3pt}\textrm {is the identity matrix}\hspace {3pt} \\ {} & {} & \Phi \left (t \right )=\phi \left (t \right )\cdot \frac {1}{\phi \left (0\right )} \\ {} & \circ & \textrm {Substitute the value of}\hspace {3pt} \phi \left (t \right )\hspace {3pt}\textrm {and}\hspace {3pt} \phi \left (0\right ) \\ {} & {} & \Phi \left (t \right )=\left [\begin {array}{ccc} \frac {{\mathrm e}^{-4 t}}{16} & {\mathrm e}^{t} & \frac {{\mathrm e}^{3 t}}{9} \\ -\frac {{\mathrm e}^{-4 t}}{4} & {\mathrm e}^{t} & \frac {{\mathrm e}^{3 t}}{3} \\ {\mathrm e}^{-4 t} & {\mathrm e}^{t} & {\mathrm e}^{3 t} \end {array}\right ]\cdot \frac {1}{\left [\begin {array}{ccc} \frac {1}{16} & 1 & \frac {1}{9} \\ -\frac {1}{4} & 1 & \frac {1}{3} \\ 1 & 1 & 1 \end {array}\right ]} \\ {} & \circ & \textrm {Evaluate and simplify to get the fundamental matrix}\hspace {3pt} \\ {} & {} & \Phi \left (t \right )=\left [\begin {array}{ccc} -\frac {\left (10 \,{\mathrm e}^{7 t}-42 \,{\mathrm e}^{5 t}-3\right ) {\mathrm e}^{-4 t}}{35} & \frac {\left (15 \,{\mathrm e}^{7 t}-7 \,{\mathrm e}^{5 t}-8\right ) {\mathrm e}^{-4 t}}{70} & \frac {\left (5 \,{\mathrm e}^{7 t}-7 \,{\mathrm e}^{5 t}+2\right ) {\mathrm e}^{-4 t}}{70} \\ -\frac {6 \left (5 \,{\mathrm e}^{7 t}-7 \,{\mathrm e}^{5 t}+2\right ) {\mathrm e}^{-4 t}}{35} & \frac {\left (45 \,{\mathrm e}^{7 t}-7 \,{\mathrm e}^{5 t}+32\right ) {\mathrm e}^{-4 t}}{70} & \frac {\left (15 \,{\mathrm e}^{7 t}-7 \,{\mathrm e}^{5 t}-8\right ) {\mathrm e}^{-4 t}}{70} \\ -\frac {6 \left (15 \,{\mathrm e}^{7 t}-7 \,{\mathrm e}^{5 t}-8\right ) {\mathrm e}^{-4 t}}{35} & \frac {\left (135 \,{\mathrm e}^{7 t}-7 \,{\mathrm e}^{5 t}-128\right ) {\mathrm e}^{-4 t}}{70} & \frac {\left (45 \,{\mathrm e}^{7 t}-7 \,{\mathrm e}^{5 t}+32\right ) {\mathrm e}^{-4 t}}{70} \end {array}\right ] \\ \square & {} & \textrm {Find a particular solution of the system of ODEs using variation of parameters}\hspace {3pt} \\ {} & \circ & \textrm {Let the particular solution be the fundamental matrix multiplied by}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (t \right )\hspace {3pt}\textrm {and solve for}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (t \right ) \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}\left (t \right )=\Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right ) \\ {} & \circ & \textrm {Take the derivative of the particular solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}^{\prime }\left (t \right )=\Phi ^{\prime }\left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right )+\Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (t \right ) \\ {} & \circ & \textrm {Substitute particular solution and its derivative into the system of ODEs}\hspace {3pt} \\ {} & {} & \Phi ^{\prime }\left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right )+\Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (t \right )=A \cdot \Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right )+{\moverset {\rightarrow }{f}}\left (t \right ) \\ {} & \circ & \textrm {The fundamental matrix has columns that are solutions to the homogeneous system so its derivative follows that of the homogeneous system}\hspace {3pt} \\ {} & {} & A \cdot \Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right )+\Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (t \right )=A \cdot \Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right )+{\moverset {\rightarrow }{f}}\left (t \right ) \\ {} & \circ & \textrm {Cancel like terms}\hspace {3pt} \\ {} & {} & \Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (t \right )={\moverset {\rightarrow }{f}}\left (t \right ) \\ {} & \circ & \textrm {Multiply by the inverse of the fundamental matrix}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{v}}^{\prime }\left (t \right )=\frac {1}{\Phi \left (t \right )}\cdot {\moverset {\rightarrow }{f}}\left (t \right ) \\ {} & \circ & \textrm {Integrate to solve for}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (t \right ) \\ {} & {} & {\moverset {\rightarrow }{v}}\left (t \right )=\int _{0}^{t}\frac {1}{\Phi \left (s \right )}\cdot {\moverset {\rightarrow }{f}}\left (s \right )d s \\ {} & \circ & \textrm {Plug}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (t \right )\hspace {3pt}\textrm {into the equation for the particular solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}\left (t \right )=\Phi \left (t \right )\cdot \left (\int _{0}^{t}\frac {1}{\Phi \left (s \right )}\cdot {\moverset {\rightarrow }{f}}\left (s \right )d s \right ) \\ {} & \circ & \textrm {Plug in the fundamental matrix and the forcing function and compute}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}\left (t \right )=\left [\begin {array}{c} \frac {\left (51 \,{\mathrm e}^{7 t}+84 \cos \left (t \right ) {\mathrm e}^{4 t}-119 \,{\mathrm e}^{5 t}-98 \,{\mathrm e}^{4 t} \sin \left (t \right )-16\right ) {\mathrm e}^{-4 t}}{2380} \\ -\frac {{\mathrm e}^{-4 t} \left (98 \cos \left (t \right ) {\mathrm e}^{4 t}+84 \,{\mathrm e}^{4 t} \sin \left (t \right )-153 \,{\mathrm e}^{7 t}+119 \,{\mathrm e}^{5 t}-64\right )}{2380} \\ -\frac {{\mathrm e}^{-4 t} \left (84 \cos \left (t \right ) {\mathrm e}^{4 t}-98 \,{\mathrm e}^{4 t} \sin \left (t \right )-459 \,{\mathrm e}^{7 t}+119 \,{\mathrm e}^{5 t}+256\right )}{2380} \end {array}\right ] \\ \bullet & {} & \textrm {Plug particular solution back into general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}\left (t \right )=c_{1} {\moverset {\rightarrow }{y}}_{1}+c_{2} {\moverset {\rightarrow }{y}}_{2}+c_{3} {\moverset {\rightarrow }{y}}_{3}+\left [\begin {array}{c} \frac {\left (51 \,{\mathrm e}^{7 t}+84 \cos \left (t \right ) {\mathrm e}^{4 t}-119 \,{\mathrm e}^{5 t}-98 \,{\mathrm e}^{4 t} \sin \left (t \right )-16\right ) {\mathrm e}^{-4 t}}{2380} \\ -\frac {{\mathrm e}^{-4 t} \left (98 \cos \left (t \right ) {\mathrm e}^{4 t}+84 \,{\mathrm e}^{4 t} \sin \left (t \right )-153 \,{\mathrm e}^{7 t}+119 \,{\mathrm e}^{5 t}-64\right )}{2380} \\ -\frac {{\mathrm e}^{-4 t} \left (84 \cos \left (t \right ) {\mathrm e}^{4 t}-98 \,{\mathrm e}^{4 t} \sin \left (t \right )-459 \,{\mathrm e}^{7 t}+119 \,{\mathrm e}^{5 t}+256\right )}{2380} \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {{\mathrm e}^{-4 t} \left (\left (6 \cos \left (t \right )-7 \sin \left (t \right )\right ) {\mathrm e}^{4 t}+\frac {170 c_{3} {\mathrm e}^{7 t}}{9}+170 c_{2} {\mathrm e}^{5 t}+\frac {51 \,{\mathrm e}^{7 t}}{14}+\frac {85 c_{1}}{8}-\frac {17 \,{\mathrm e}^{5 t}}{2}-\frac {8}{7}\right )}{170} \end {array} \]

Maple trace

`Methods for third order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 3; linear nonhomogeneous with symmetry [0,1] 
trying high order linear exact nonhomogeneous 
trying differential order: 3; missing the dependent variable 
checking if the LODE has constant coefficients 
<- constant coefficients successful`
 

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 37

dsolve(diff(y(t),t$3)-13*diff(y(t),t)+12*y(t)=cos(t),y(t), singsol=all)
 

\[ y \left (t \right ) = \left (\left (\frac {3 \cos \left (t \right )}{85}-\frac {7 \sin \left (t \right )}{170}\right ) {\mathrm e}^{4 t}+c_{3} {\mathrm e}^{7 t}+c_{1} {\mathrm e}^{5 t}+c_{2} \right ) {\mathrm e}^{-4 t} \]

✓ Solution by Mathematica

Time used: 0.033 (sec). Leaf size: 40

DSolve[y'''[t]-13*y'[t]+12*y[t]==Cos[t],y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -\frac {7 \sin (t)}{170}+\frac {3 \cos (t)}{85}+c_1 e^{-4 t}+c_2 e^t+c_3 e^{3 t} \]