7.8 problem Exercise 20.9, page 220

Internal problem ID [4579]
Internal file name [OUTPUT/4072_Sunday_June_05_2022_12_18_43_PM_61299716/index.tex]

Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 4. Higher order linear differential equations. Lesson 20. Constant coefficients
Problem number: Exercise 20.9, page 220.
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]]

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

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

Summary

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

Verification of solutions

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

7.8.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+y^{\prime \prime }-4 y^{\prime }-2 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 )=-4 y_{4}\left (x \right )-y_{3}\left (x \right )+4 y_{2}\left (x \right )+2 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 )=-4 y_{4}\left (x \right )-y_{3}\left (x \right )+4 y_{2}\left (x \right )+2 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 \\ 2 & 4 & -1 & -4 \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 \\ 2 & 4 & -1 & -4 \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 [1, \left [\begin {array}{c} 1 \\ 1 \\ 1 \\ 1 \end {array}\right ]\right ], \left [-\sqrt {2}-2, \left [\begin {array}{c} \frac {1}{\left (-\sqrt {2}-2\right )^{3}} \\ \frac {1}{\left (-\sqrt {2}-2\right )^{2}} \\ \frac {1}{-\sqrt {2}-2} \\ 1 \end {array}\right ]\right ], \left [\sqrt {2}-2, \left [\begin {array}{c} \frac {1}{\left (\sqrt {2}-2\right )^{3}} \\ \frac {1}{\left (\sqrt {2}-2\right )^{2}} \\ \frac {1}{\sqrt {2}-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 [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}}_{2}={\mathrm e}^{x}\cdot \left [\begin {array}{c} 1 \\ 1 \\ 1 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\sqrt {2}-2, \left [\begin {array}{c} \frac {1}{\left (-\sqrt {2}-2\right )^{3}} \\ \frac {1}{\left (-\sqrt {2}-2\right )^{2}} \\ \frac {1}{-\sqrt {2}-2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{3}={\mathrm e}^{\left (-\sqrt {2}-2\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-\sqrt {2}-2\right )^{3}} \\ \frac {1}{\left (-\sqrt {2}-2\right )^{2}} \\ \frac {1}{-\sqrt {2}-2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [\sqrt {2}-2, \left [\begin {array}{c} \frac {1}{\left (\sqrt {2}-2\right )^{3}} \\ \frac {1}{\left (\sqrt {2}-2\right )^{2}} \\ \frac {1}{\sqrt {2}-2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{4}={\mathrm e}^{\left (\sqrt {2}-2\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (\sqrt {2}-2\right )^{3}} \\ \frac {1}{\left (\sqrt {2}-2\right )^{2}} \\ \frac {1}{\sqrt {2}-2} \\ 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}^{-x}\cdot \left [\begin {array}{c} -1 \\ 1 \\ -1 \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{x}\cdot \left [\begin {array}{c} 1 \\ 1 \\ 1 \\ 1 \end {array}\right ]+{\mathrm e}^{\left (-\sqrt {2}-2\right ) x} c_{3} \cdot \left [\begin {array}{c} \frac {1}{\left (-\sqrt {2}-2\right )^{3}} \\ \frac {1}{\left (-\sqrt {2}-2\right )^{2}} \\ \frac {1}{-\sqrt {2}-2} \\ 1 \end {array}\right ]+{\mathrm e}^{\left (\sqrt {2}-2\right ) x} c_{4} \cdot \left [\begin {array}{c} \frac {1}{\left (\sqrt {2}-2\right )^{3}} \\ \frac {1}{\left (\sqrt {2}-2\right )^{2}} \\ \frac {1}{\sqrt {2}-2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {c_{3} \left (7 \sqrt {2}-10\right ) {\mathrm e}^{-\left (2+\sqrt {2}\right ) x}}{4}+\frac {c_{4} \left (-7 \sqrt {2}-10\right ) {\mathrm e}^{\left (\sqrt {2}-2\right ) x}}{4}-c_{1} {\mathrm e}^{-x}+c_{2} {\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`
 

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 36

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 49

DSolve[y''''[x]+4*y'''[x]+y''[x]-4*y'[x]-2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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