problem Problem 3(d)

Internal problem ID [12246]
Internal ﬁle name [OUTPUT/10899_Thursday_September_28_2023_01_08_25_AM_88166950/index.tex]

Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section: Chapter 4, Second and Higher Order Linear Diﬀerential Equations. Problems page 221
Problem number: Problem 3(d).
ODE order: 4.
ODE degree: 1.

The type(s) of ODE detected by this program : "higher_order_linear_constant_coeﬃcients_ODE"

\[ \boxed {3 y^{\prime \prime \prime \prime }-2 y^{\prime \prime }+y^{\prime }=0} \] The characteristic equation is \[ 3 \lambda ^{4}-2 \lambda ^{2}+\lambda = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= -1\\ \lambda _3 &= \frac {i \sqrt {3}}{6}+\frac {1}{2}\\ \lambda _4 &= \frac {1}{2}-\frac {i \sqrt {3}}{6} \end {align*}

Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{-x} c_{1} +c_{2} +{\mathrm e}^{\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}\right ) x} c_{3} +{\mathrm e}^{\left (\frac {i \sqrt {3}}{6}+\frac {1}{2}\right ) x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-x}\\ y_2 &= 1\\ y_3 &= {\mathrm e}^{\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}\right ) x}\\ y_4 &= {\mathrm e}^{\left (\frac {i \sqrt {3}}{6}+\frac {1}{2}\right ) x} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-x} c_{1} +c_{2} +{\mathrm e}^{\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}\right ) x} c_{3} +{\mathrm e}^{\left (\frac {i \sqrt {3}}{6}+\frac {1}{2}\right ) x} c_{4} \\ \end{align*}

Veriﬁcation of solutions

\[ y = {\mathrm e}^{-x} c_{1} +c_{2} +{\mathrm e}^{\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}\right ) x} c_{3} +{\mathrm e}^{\left (\frac {i \sqrt {3}}{6}+\frac {1}{2}\right ) x} c_{4} \] Veriﬁed OK.

2.26.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 3 y^{\prime \prime \prime \prime }-2 y^{\prime \prime }+y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & y^{\prime \prime \prime \prime } \\ \bullet & {} & \textrm {Isolate 4th derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime \prime }=\frac {2 y^{\prime \prime }}{3}-\frac {y^{\prime }}{3} \\ \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 \prime }-\frac {2 y^{\prime \prime }}{3}+\frac {y^{\prime }}{3}=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 (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 {Isolate for}\hspace {3pt} y_{4}^{\prime }\left (x \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{4}^{\prime }\left (x \right )=\frac {2 y_{3}\left (x \right )}{3}-\frac {y_{2}\left (x \right )}{3} \\ & {} & \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_{4}^{\prime }\left (x \right )=\frac {2 y_{3}\left (x \right )}{3}-\frac {y_{2}\left (x \right )}{3}\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 ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & -\frac {1}{3} & \frac {2}{3} & 0 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \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 \\ 0 & -\frac {1}{3} & \frac {2}{3} & 0 \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 [-1, \left [\begin {array}{c} -1 \\ 1 \\ -1 \\ 1 \end {array}\right ]\right ], \left [0, \left [\begin {array}{c} 1 \\ 0 \\ 0 \\ 0 \end {array}\right ]\right ], \left [\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}, \left [\begin {array}{c} \frac {1}{\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}\right )^{3}} \\ \frac {1}{\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}\right )^{2}} \\ \frac {1}{\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}} \\ 1 \end {array}\right ]\right ], \left [\frac {\mathrm {I} \sqrt {3}}{6}+\frac {1}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\mathrm {I} \sqrt {3}}{6}+\frac {1}{2}\right )^{3}} \\ \frac {1}{\left (\frac {\mathrm {I} \sqrt {3}}{6}+\frac {1}{2}\right )^{2}} \\ \frac {1}{\frac {\mathrm {I} \sqrt {3}}{6}+\frac {1}{2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-1, \left [\begin {array}{c} -1 \\ 1 \\ -1 \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{-x}\cdot \left [\begin {array}{c} -1 \\ 1 \\ -1 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [0, \left [\begin {array}{c} 1 \\ 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} 1 \\ 0 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}, \left [\begin {array}{c} \frac {1}{\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}\right )^{3}} \\ \frac {1}{\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}\right )^{2}} \\ \frac {1}{\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}\right )^{3}} \\ \frac {1}{\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}\right )^{2}} \\ \frac {1}{\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}} \\ 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}^{\frac {x}{2}}\cdot \left (\cos \left (\frac {\sqrt {3}\, x}{6}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, x}{6}\right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}\right )^{3}} \\ \frac {1}{\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}\right )^{2}} \\ \frac {1}{\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\frac {x}{2}}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {3}\, x}{6}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, x}{6}\right )}{\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}\right )^{3}} \\ \frac {\cos \left (\frac {\sqrt {3}\, x}{6}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, x}{6}\right )}{\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}\right )^{2}} \\ \frac {\cos \left (\frac {\sqrt {3}\, x}{6}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, x}{6}\right )}{\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{6}} \\ \cos \left (\frac {\sqrt {3}\, x}{6}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, x}{6}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{3}\left (x \right )={\mathrm e}^{\frac {x}{2}}\cdot \left [\begin {array}{c} 3 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{6}\right ) \\ \frac {3 \cos \left (\frac {\sqrt {3}\, x}{6}\right )}{2}+\frac {3 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{6}\right )}{2} \\ \frac {3 \cos \left (\frac {\sqrt {3}\, x}{6}\right )}{2}+\frac {\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{6}\right )}{2} \\ \cos \left (\frac {\sqrt {3}\, x}{6}\right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{4}\left (x \right )={\mathrm e}^{\frac {x}{2}}\cdot \left [\begin {array}{c} 3 \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, x}{6}\right ) \\ \frac {3 \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, x}{6}\right )}{2}-\frac {3 \sin \left (\frac {\sqrt {3}\, x}{6}\right )}{2} \\ \frac {\sqrt {3}\, \cos \left (\frac {\sqrt {3}\, x}{6}\right )}{2}-\frac {3 \sin \left (\frac {\sqrt {3}\, x}{6}\right )}{2} \\ -\sin \left (\frac {\sqrt {3}\, x}{6}\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}\left (x \right )+c_{4} {\moverset {\rightarrow }{y}}_{4}\left (x \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}={\mathrm e}^{-x} c_{1} \cdot \left [\begin {array}{c} -1 \\ 1 \\ -1 \\ 1 \end {array}\right ]+c_{3} {\mathrm e}^{\frac {x}{2}}\cdot \left [\begin {array}{c} 3 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{6}\right ) \\ \frac {3 \cos \left (\frac {\sqrt {3}\, x}{6}\right )}{2}+\frac {3 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{6}\right )}{2} \\ \frac {3 \cos \left (\frac {\sqrt {3}\, x}{6}\right )}{2}+\frac {\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{6}\right )}{2} \\ \cos \left (\frac {\sqrt {3}\, x}{6}\right ) \end {array}\right ]+c_{4} {\mathrm e}^{\frac {x}{2}}\cdot \left [\begin {array}{c} 3 \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, x}{6}\right ) \\ \frac {3 \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, x}{6}\right )}{2}-\frac {3 \sin \left (\frac {\sqrt {3}\, x}{6}\right )}{2} \\ \frac {\sqrt {3}\, \cos \left (\frac {\sqrt {3}\, x}{6}\right )}{2}-\frac {3 \sin \left (\frac {\sqrt {3}\, x}{6}\right )}{2} \\ -\sin \left (\frac {\sqrt {3}\, x}{6}\right ) \end {array}\right ]+\left [\begin {array}{c} c_{2} \\ 0 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\left (3 c_{3} {\mathrm e}^{\frac {3 x}{2}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{6}\right )+3 c_{4} {\mathrm e}^{\frac {3 x}{2}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, x}{6}\right )+{\mathrm e}^{x} c_{2} -c_{1} \right ) {\mathrm e}^{-x} \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`

\[ y \left (x \right ) = \left (c_{3} {\mathrm e}^{\frac {3 x}{2}} \sin \left (\frac {\sqrt {3}\, x}{6}\right )+c_{4} {\mathrm e}^{\frac {3 x}{2}} \cos \left (\frac {\sqrt {3}\, x}{6}\right )+c_{1} {\mathrm e}^{x}+c_{2} \right ) {\mathrm e}^{-x} \]

\[ y(x)\to c_3 \left (-e^{-x}\right )-\frac {1}{2} \left (\sqrt {3} c_1-3 c_2\right ) e^{x/2} \cos \left (\frac {x}{2 \sqrt {3}}\right )+\frac {1}{2} \left (3 c_1+\sqrt {3} c_2\right ) e^{x/2} \sin \left (\frac {x}{2 \sqrt {3}}\right )+c_4 \]

2.26 problem Problem 3(d)

2.26.1 Maple step by step solution