17.13 problem 13

Internal problem ID [13810]
Internal file name [OUTPUT/12982_Sunday_February_18_2024_08_02_29_AM_372947/index.tex]

Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section: Chapter 25. Review exercises for part III. page 447
Problem number: 13.
ODE order: 5.
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]]

\[ \boxed {y^{\left (5\right )}-6 y^{\prime \prime \prime \prime }+13 y^{\prime \prime \prime }=0} \] The characteristic equation is \[ \lambda ^{5}-6 \lambda ^{4}+13 \lambda ^{3} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 0\\ \lambda _3 &= 0\\ \lambda _4 &= 3+2 i\\ \lambda _5 &= 3-2 i \end {align*}

Therefore the homogeneous solution is \[ y_h(x)=c_{3} x^{2}+c_{2} x +c_{1} +{\mathrm e}^{\left (3+2 i\right ) x} c_{4} +{\mathrm e}^{\left (3-2 i\right ) x} c_{5} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= 1\\ y_2 &= x\\ y_3 &= x^{2}\\ y_4 &= {\mathrm e}^{\left (3+2 i\right ) x}\\ y_5 &= {\mathrm e}^{\left (3-2 i\right ) x} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= c_{3} x^{2}+c_{2} x +c_{1} +{\mathrm e}^{\left (3+2 i\right ) x} c_{4} +{\mathrm e}^{\left (3-2 i\right ) x} c_{5} \\ \end{align*}

Verification of solutions

\[ y = c_{3} x^{2}+c_{2} x +c_{1} +{\mathrm e}^{\left (3+2 i\right ) x} c_{4} +{\mathrm e}^{\left (3-2 i\right ) x} c_{5} \] Verified OK.

17.13.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\left (5\right )}-6 y^{\prime \prime \prime \prime }+13 y^{\prime \prime \prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 5 \\ {} & {} & y^{\left (5\right )} \\ \square & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{1}\left (x \right ) \\ {} & {} & y_{1}\left (x \right )=y \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{2}\left (x \right ) \\ {} & {} & y_{2}\left (x \right )=y^{\prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{3}\left (x \right ) \\ {} & {} & y_{3}\left (x \right )=y^{\prime \prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{4}\left (x \right ) \\ {} & {} & y_{4}\left (x \right )=y^{\prime \prime \prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{5}\left (x \right ) \\ {} & {} & y_{5}\left (x \right )=y^{\prime \prime \prime \prime } \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} y_{5}^{\prime }\left (x \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{5}^{\prime }\left (x \right )=6 y_{5}\left (x \right )-13 y_{4}\left (x \right ) \\ & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & {} & \left [y_{2}\left (x \right )=y_{1}^{\prime }\left (x \right ), y_{3}\left (x \right )=y_{2}^{\prime }\left (x \right ), y_{4}\left (x \right )=y_{3}^{\prime }\left (x \right ), y_{5}\left (x \right )=y_{4}^{\prime }\left (x \right ), y_{5}^{\prime }\left (x \right )=6 y_{5}\left (x \right )-13 y_{4}\left (x \right )\right ] \\ \bullet & {} & \textrm {Define vector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}\left (x \right )=\left [\begin {array}{c} y_{1}\left (x \right ) \\ y_{2}\left (x \right ) \\ y_{3}\left (x \right ) \\ y_{4}\left (x \right ) \\ y_{5}\left (x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=\left [\begin {array}{ccccc} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & -13 & 6 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccccc} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & -13 & 6 \end {array}\right ] \\ \bullet & {} & \textrm {Rewrite the system as}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=A \cdot {\moverset {\rightarrow }{y}}\left (x \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 [0, \left [\begin {array}{c} 1 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ]\right ], \left [0, \left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ]\right ], \left [0, \left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ]\right ], \left [3-2 \,\mathrm {I}, \left [\begin {array}{c} -\frac {119}{28561}+\frac {120 \,\mathrm {I}}{28561} \\ -\frac {9}{2197}+\frac {46 \,\mathrm {I}}{2197} \\ \frac {5}{169}+\frac {12 \,\mathrm {I}}{169} \\ \frac {3}{13}+\frac {2 \,\mathrm {I}}{13} \\ 1 \end {array}\right ]\right ], \left [3+2 \,\mathrm {I}, \left [\begin {array}{c} -\frac {119}{28561}-\frac {120 \,\mathrm {I}}{28561} \\ -\frac {9}{2197}-\frac {46 \,\mathrm {I}}{2197} \\ \frac {5}{169}-\frac {12 \,\mathrm {I}}{169} \\ \frac {3}{13}-\frac {2 \,\mathrm {I}}{13} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [0, \left [\begin {array}{c} 1 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}=\left [\begin {array}{c} 1 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [0, \left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{2}=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [0, \left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{3}=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [3-2 \,\mathrm {I}, \left [\begin {array}{c} -\frac {119}{28561}+\frac {120 \,\mathrm {I}}{28561} \\ -\frac {9}{2197}+\frac {46 \,\mathrm {I}}{2197} \\ \frac {5}{169}+\frac {12 \,\mathrm {I}}{169} \\ \frac {3}{13}+\frac {2 \,\mathrm {I}}{13} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (3-2 \,\mathrm {I}\right ) x}\cdot \left [\begin {array}{c} -\frac {119}{28561}+\frac {120 \,\mathrm {I}}{28561} \\ -\frac {9}{2197}+\frac {46 \,\mathrm {I}}{2197} \\ \frac {5}{169}+\frac {12 \,\mathrm {I}}{169} \\ \frac {3}{13}+\frac {2 \,\mathrm {I}}{13} \\ 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}^{3 x}\cdot \left (\cos \left (2 x \right )-\mathrm {I} \sin \left (2 x \right )\right )\cdot \left [\begin {array}{c} -\frac {119}{28561}+\frac {120 \,\mathrm {I}}{28561} \\ -\frac {9}{2197}+\frac {46 \,\mathrm {I}}{2197} \\ \frac {5}{169}+\frac {12 \,\mathrm {I}}{169} \\ \frac {3}{13}+\frac {2 \,\mathrm {I}}{13} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{3 x}\cdot \left [\begin {array}{c} \left (-\frac {119}{28561}+\frac {120 \,\mathrm {I}}{28561}\right ) \left (\cos \left (2 x \right )-\mathrm {I} \sin \left (2 x \right )\right ) \\ \left (-\frac {9}{2197}+\frac {46 \,\mathrm {I}}{2197}\right ) \left (\cos \left (2 x \right )-\mathrm {I} \sin \left (2 x \right )\right ) \\ \left (\frac {5}{169}+\frac {12 \,\mathrm {I}}{169}\right ) \left (\cos \left (2 x \right )-\mathrm {I} \sin \left (2 x \right )\right ) \\ \left (\frac {3}{13}+\frac {2 \,\mathrm {I}}{13}\right ) \left (\cos \left (2 x \right )-\mathrm {I} \sin \left (2 x \right )\right ) \\ \cos \left (2 x \right )-\mathrm {I} \sin \left (2 x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{4}\left (x \right )={\mathrm e}^{3 x}\cdot \left [\begin {array}{c} -\frac {119 \cos \left (2 x \right )}{28561}+\frac {120 \sin \left (2 x \right )}{28561} \\ -\frac {9 \cos \left (2 x \right )}{2197}+\frac {46 \sin \left (2 x \right )}{2197} \\ \frac {5 \cos \left (2 x \right )}{169}+\frac {12 \sin \left (2 x \right )}{169} \\ \frac {3 \cos \left (2 x \right )}{13}+\frac {2 \sin \left (2 x \right )}{13} \\ \cos \left (2 x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{5}\left (x \right )={\mathrm e}^{3 x}\cdot \left [\begin {array}{c} \frac {119 \sin \left (2 x \right )}{28561}+\frac {120 \cos \left (2 x \right )}{28561} \\ \frac {9 \sin \left (2 x \right )}{2197}+\frac {46 \cos \left (2 x \right )}{2197} \\ -\frac {5 \sin \left (2 x \right )}{169}+\frac {12 \cos \left (2 x \right )}{169} \\ -\frac {3 \sin \left (2 x \right )}{13}+\frac {2 \cos \left (2 x \right )}{13} \\ -\sin \left (2 x \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}+c_{3} {\moverset {\rightarrow }{y}}_{3}+c_{4} {\moverset {\rightarrow }{y}}_{4}\left (x \right )+c_{5} {\moverset {\rightarrow }{y}}_{5}\left (x \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}={\mathrm e}^{3 x} c_{4} \cdot \left [\begin {array}{c} -\frac {119 \cos \left (2 x \right )}{28561}+\frac {120 \sin \left (2 x \right )}{28561} \\ -\frac {9 \cos \left (2 x \right )}{2197}+\frac {46 \sin \left (2 x \right )}{2197} \\ \frac {5 \cos \left (2 x \right )}{169}+\frac {12 \sin \left (2 x \right )}{169} \\ \frac {3 \cos \left (2 x \right )}{13}+\frac {2 \sin \left (2 x \right )}{13} \\ \cos \left (2 x \right ) \end {array}\right ]+c_{5} {\mathrm e}^{3 x}\cdot \left [\begin {array}{c} \frac {119 \sin \left (2 x \right )}{28561}+\frac {120 \cos \left (2 x \right )}{28561} \\ \frac {9 \sin \left (2 x \right )}{2197}+\frac {46 \cos \left (2 x \right )}{2197} \\ -\frac {5 \sin \left (2 x \right )}{169}+\frac {12 \cos \left (2 x \right )}{169} \\ -\frac {3 \sin \left (2 x \right )}{13}+\frac {2 \cos \left (2 x \right )}{13} \\ -\sin \left (2 x \right ) \end {array}\right ]+\left [\begin {array}{c} c_{1} \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {\left (\left (-119 c_{4} +120 c_{5} \right ) \cos \left (2 x \right )+120 \left (c_{4} +\frac {119 c_{5}}{120}\right ) \sin \left (2 x \right )\right ) {\mathrm e}^{3 x}}{28561}+c_{1} \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: 0.0 (sec). Leaf size: 34

dsolve(diff(y(x),x$5)-6*diff(y(x),x$4)+13*diff(y(x),x$3)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = c_{1} +c_{2} x +x^{2} c_{3} +c_{4} {\mathrm e}^{3 x} \sin \left (2 x \right )+c_{5} {\mathrm e}^{3 x} \cos \left (2 x \right ) \]

✓ Solution by Mathematica

Time used: 2.104 (sec). Leaf size: 56

DSolve[y'''''[x]-6*y''''[x]+13*y'''[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_5 x^2+c_4 x-\frac {e^{3 x} ((46 c_1+9 c_2) \cos (2 x)+(9 c_1-46 c_2) \sin (2 x))}{2197}+c_3 \]