2.11 problem 18

Internal problem ID [832]
Internal file name [OUTPUT/832_Sunday_June_05_2022_01_50_44_AM_67612248/index.tex]

Book: Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section: Chapter 4.2, Higher order linear differential equations. Constant coefficients. page 180
Problem number: 18.
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]]

\[ \boxed {y^{\prime \prime \prime \prime }-7 y^{\prime \prime \prime }+6 y^{\prime \prime }+30 y^{\prime }-36 y=0} \] The characteristic equation is \[ \lambda ^{4}-7 \lambda ^{3}+6 \lambda ^{2}+30 \lambda -36 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 3\\ \lambda _2 &= -2\\ \lambda _3 &= 3-\sqrt {3}\\ \lambda _4 &= 3+\sqrt {3} \end {align*}

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

Summary

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

Verification of solutions

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

2.11.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime \prime }-7 y^{\prime \prime \prime }+6 y^{\prime \prime }+30 y^{\prime }-36 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 (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 )=7 y_{4}\left (x \right )-6 y_{3}\left (x \right )-30 y_{2}\left (x \right )+36 y_{1}\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_{4}^{\prime }\left (x \right )=7 y_{4}\left (x \right )-6 y_{3}\left (x \right )-30 y_{2}\left (x \right )+36 y_{1}\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 ) \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 \\ 36 & -30 & -6 & 7 \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 \\ 36 & -30 & -6 & 7 \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 [-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 [3-\sqrt {3}, \left [\begin {array}{c} \frac {1}{\left (3-\sqrt {3}\right )^{3}} \\ \frac {1}{\left (3-\sqrt {3}\right )^{2}} \\ \frac {1}{3-\sqrt {3}} \\ 1 \end {array}\right ]\right ], \left [3+\sqrt {3}, \left [\begin {array}{c} \frac {1}{\left (3+\sqrt {3}\right )^{3}} \\ \frac {1}{\left (3+\sqrt {3}\right )^{2}} \\ \frac {1}{3+\sqrt {3}} \\ 1 \end {array}\right ]\right ]\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}}_{1}={\mathrm e}^{-2 x}\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}}_{2}={\mathrm e}^{3 x}\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 [3-\sqrt {3}, \left [\begin {array}{c} \frac {1}{\left (3-\sqrt {3}\right )^{3}} \\ \frac {1}{\left (3-\sqrt {3}\right )^{2}} \\ \frac {1}{3-\sqrt {3}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{3}={\mathrm e}^{\left (3-\sqrt {3}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (3-\sqrt {3}\right )^{3}} \\ \frac {1}{\left (3-\sqrt {3}\right )^{2}} \\ \frac {1}{3-\sqrt {3}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [3+\sqrt {3}, \left [\begin {array}{c} \frac {1}{\left (3+\sqrt {3}\right )^{3}} \\ \frac {1}{\left (3+\sqrt {3}\right )^{2}} \\ \frac {1}{3+\sqrt {3}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{4}={\mathrm e}^{\left (3+\sqrt {3}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (3+\sqrt {3}\right )^{3}} \\ \frac {1}{\left (3+\sqrt {3}\right )^{2}} \\ \frac {1}{3+\sqrt {3}} \\ 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}^{-2 x}\cdot \left [\begin {array}{c} -\frac {1}{8} \\ \frac {1}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{3 x}\cdot \left [\begin {array}{c} \frac {1}{27} \\ \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]+c_{3} {\mathrm e}^{\left (3-\sqrt {3}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (3-\sqrt {3}\right )^{3}} \\ \frac {1}{\left (3-\sqrt {3}\right )^{2}} \\ \frac {1}{3-\sqrt {3}} \\ 1 \end {array}\right ]+c_{4} {\mathrm e}^{\left (3+\sqrt {3}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (3+\sqrt {3}\right )^{3}} \\ \frac {1}{\left (3+\sqrt {3}\right )^{2}} \\ \frac {1}{3+\sqrt {3}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {5 \,{\mathrm e}^{-2 x} \left (c_{3} \left (\sqrt {3}+\frac {9}{5}\right ) {\mathrm e}^{-x \left (-5+\sqrt {3}\right )}-\left (\sqrt {3}-\frac {9}{5}\right ) c_{4} {\mathrm e}^{x \left (5+\sqrt {3}\right )}+\frac {4 c_{2} {\mathrm e}^{5 x}}{15}-\frac {9 c_{1}}{10}\right )}{36} \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: 0.015 (sec). Leaf size: 38

dsolve(diff(y(x),x$4)-7*diff(y(x),x$3)+6*diff(y(x),x$2)+30*diff(y(x),x)-36*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 51

DSolve[y''''[x]-7*y'''[x]+6*y''[x]+30*y'[x]-36*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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