2.8 problem problem 17

Internal problem ID [942]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 5.3, Higher-Order Linear Differential Equations. Homogeneous Equations with Constant Coefficients. Page 300
Problem number : problem 17
Date solved : Thursday, October 17, 2024 at 01:25:20 AM
CAS classification : [[_high_order, _missing_x]]

Solve

\begin{align*} 6 y^{\prime \prime \prime \prime }+11 y^{\prime \prime }+4 y&=0 \end{align*}

2.8.1 Solved as higher order constant coeff ode

Time used: 0.065 (sec)

The characteristic equation is

\[ 6 \lambda ^{4}+11 \lambda ^{2}+4 = 0 \]

The roots of the above equation are

\begin{align*} \lambda _1 &= \frac {i \sqrt {2}}{2}\\ \lambda _2 &= -\frac {i \sqrt {2}}{2}\\ \lambda _3 &= \frac {2 i \sqrt {3}}{3}\\ \lambda _4 &= -\frac {2 i \sqrt {3}}{3} \end{align*}

Therefore the homogeneous solution is

\[ y_h(x)={\mathrm e}^{-\frac {i \sqrt {2}\, x}{2}} c_1 +{\mathrm e}^{\frac {2 i \sqrt {3}\, x}{3}} c_2 +{\mathrm e}^{\frac {i \sqrt {2}\, x}{2}} c_3 +{\mathrm e}^{-\frac {2 i \sqrt {3}\, x}{3}} c_4 \]

The fundamental set of solutions for the homogeneous solution are the following

\begin{align*} y_1 &= {\mathrm e}^{-\frac {i \sqrt {2}\, x}{2}}\\ y_2 &= {\mathrm e}^{\frac {2 i \sqrt {3}\, x}{3}}\\ y_3 &= {\mathrm e}^{\frac {i \sqrt {2}\, x}{2}}\\ y_4 &= {\mathrm e}^{-\frac {2 i \sqrt {3}\, x}{3}} \end{align*}

2.8.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 6 \frac {d^{4}}{d x^{4}}y \left (x \right )+11 \frac {d^{2}}{d x^{2}}y \left (x \right )+4 y \left (x \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & \frac {d^{4}}{d x^{4}}y \left (x \right ) \\ \bullet & {} & \textrm {Isolate 4th derivative}\hspace {3pt} \\ {} & {} & \frac {d^{4}}{d x^{4}}y \left (x \right )=-\frac {11 \frac {d^{2}}{d x^{2}}y \left (x \right )}{6}-\frac {2 y \left (x \right )}{3} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y \left (x \right )\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & \frac {d^{4}}{d x^{4}}y \left (x \right )+\frac {11 \frac {d^{2}}{d x^{2}}y \left (x \right )}{6}+\frac {2 y \left (x \right )}{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 \left (x \right ) \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{2}\left (x \right ) \\ {} & {} & y_{2}\left (x \right )=\frac {d}{d x}y \left (x \right ) \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{3}\left (x \right ) \\ {} & {} & y_{3}\left (x \right )=\frac {d^{2}}{d x^{2}}y \left (x \right ) \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{4}\left (x \right ) \\ {} & {} & y_{4}\left (x \right )=\frac {d^{3}}{d x^{3}}y \left (x \right ) \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} \frac {d}{d x}y_{4}\left (x \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y_{4}\left (x \right )=-\frac {11 y_{3}\left (x \right )}{6}-\frac {2 y_{1}\left (x \right )}{3} \\ & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & {} & \left [y_{2}\left (x \right )=\frac {d}{d x}y_{1}\left (x \right ), y_{3}\left (x \right )=\frac {d}{d x}y_{2}\left (x \right ), y_{4}\left (x \right )=\frac {d}{d x}y_{3}\left (x \right ), \frac {d}{d x}y_{4}\left (x \right )=-\frac {11 y_{3}\left (x \right )}{6}-\frac {2 y_{1}\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} \\ {} & {} & \frac {d}{d x}{\moverset {\rightarrow }{y}}\left (x \right )=\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -\frac {2}{3} & 0 & -\frac {11}{6} & 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 \\ -\frac {2}{3} & 0 & -\frac {11}{6} & 0 \end {array}\right ] \\ \bullet & {} & \textrm {Rewrite the system as}\hspace {3pt} \\ {} & {} & \frac {d}{d x}{\moverset {\rightarrow }{y}}\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 [-\frac {2 \,\mathrm {I}}{3} \sqrt {3}, \left [\begin {array}{c} -\frac {3 \,\mathrm {I}}{8} \sqrt {3} \\ -\frac {3}{4} \\ \frac {\mathrm {I}}{2} \sqrt {3} \\ 1 \end {array}\right ]\right ], \left [-\frac {\mathrm {I}}{2} \sqrt {2}, \left [\begin {array}{c} -2 \,\mathrm {I} \sqrt {2} \\ -2 \\ \mathrm {I} \sqrt {2} \\ 1 \end {array}\right ]\right ], \left [\frac {\mathrm {I}}{2} \sqrt {2}, \left [\begin {array}{c} 2 \,\mathrm {I} \sqrt {2} \\ -2 \\ \mathrm {-I} \sqrt {2} \\ 1 \end {array}\right ]\right ], \left [\frac {2 \,\mathrm {I}}{3} \sqrt {3}, \left [\begin {array}{c} \frac {3 \,\mathrm {I}}{8} \sqrt {3} \\ -\frac {3}{4} \\ -\frac {\mathrm {I}}{2} \sqrt {3} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-\frac {2 \,\mathrm {I}}{3} \sqrt {3}, \left [\begin {array}{c} -\frac {3 \,\mathrm {I}}{8} \sqrt {3} \\ -\frac {3}{4} \\ \frac {\mathrm {I}}{2} \sqrt {3} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-\frac {2 \,\mathrm {I}}{3} \sqrt {3}\, x}\cdot \left [\begin {array}{c} -\frac {3 \,\mathrm {I}}{8} \sqrt {3} \\ -\frac {3}{4} \\ \frac {\mathrm {I}}{2} \sqrt {3} \\ 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 \\ {} & {} & \left (\cos \left (\frac {2 \sqrt {3}\, x}{3}\right )-\mathrm {I} \sin \left (\frac {2 \sqrt {3}\, x}{3}\right )\right )\cdot \left [\begin {array}{c} -\frac {3 \,\mathrm {I}}{8} \sqrt {3} \\ -\frac {3}{4} \\ \frac {\mathrm {I}}{2} \sqrt {3} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & \left [\begin {array}{c} -\frac {3 \,\mathrm {I}}{8} \left (\cos \left (\frac {2 \sqrt {3}\, x}{3}\right )-\mathrm {I} \sin \left (\frac {2 \sqrt {3}\, x}{3}\right )\right ) \sqrt {3} \\ -\frac {3 \cos \left (\frac {2 \sqrt {3}\, x}{3}\right )}{4}+\frac {3 \,\mathrm {I} \sin \left (\frac {2 \sqrt {3}\, x}{3}\right )}{4} \\ \frac {\mathrm {I}}{2} \left (\cos \left (\frac {2 \sqrt {3}\, x}{3}\right )-\mathrm {I} \sin \left (\frac {2 \sqrt {3}\, x}{3}\right )\right ) \sqrt {3} \\ \cos \left (\frac {2 \sqrt {3}\, x}{3}\right )-\mathrm {I} \sin \left (\frac {2 \sqrt {3}\, x}{3}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{1}\left (x \right )=\left [\begin {array}{c} -\frac {3 \sqrt {3}\, \sin \left (\frac {2 \sqrt {3}\, x}{3}\right )}{8} \\ -\frac {3 \cos \left (\frac {2 \sqrt {3}\, x}{3}\right )}{4} \\ \frac {\sqrt {3}\, \sin \left (\frac {2 \sqrt {3}\, x}{3}\right )}{2} \\ \cos \left (\frac {2 \sqrt {3}\, x}{3}\right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{2}\left (x \right )=\left [\begin {array}{c} -\frac {3 \sqrt {3}\, \cos \left (\frac {2 \sqrt {3}\, x}{3}\right )}{8} \\ \frac {3 \sin \left (\frac {2 \sqrt {3}\, x}{3}\right )}{4} \\ \frac {\sqrt {3}\, \cos \left (\frac {2 \sqrt {3}\, x}{3}\right )}{2} \\ -\sin \left (\frac {2 \sqrt {3}\, x}{3}\right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-\frac {\mathrm {I}}{2} \sqrt {2}, \left [\begin {array}{c} -2 \,\mathrm {I} \sqrt {2} \\ -2 \\ \mathrm {I} \sqrt {2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-\frac {\mathrm {I}}{2} \sqrt {2}\, x}\cdot \left [\begin {array}{c} -2 \,\mathrm {I} \sqrt {2} \\ -2 \\ \mathrm {I} \sqrt {2} \\ 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 \\ {} & {} & \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )\cdot \left [\begin {array}{c} -2 \,\mathrm {I} \sqrt {2} \\ -2 \\ \mathrm {I} \sqrt {2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & \left [\begin {array}{c} -2 \,\mathrm {I} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) \sqrt {2} \\ -2 \cos \left (\frac {\sqrt {2}\, x}{2}\right )+2 \,\mathrm {I} \sin \left (\frac {\sqrt {2}\, x}{2}\right ) \\ \mathrm {I} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) \sqrt {2} \\ \cos \left (\frac {\sqrt {2}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, x}{2}\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 )=\left [\begin {array}{c} -2 \sqrt {2}\, \sin \left (\frac {\sqrt {2}\, x}{2}\right ) \\ -2 \cos \left (\frac {\sqrt {2}\, x}{2}\right ) \\ \sqrt {2}\, \sin \left (\frac {\sqrt {2}\, x}{2}\right ) \\ \cos \left (\frac {\sqrt {2}\, x}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{4}\left (x \right )=\left [\begin {array}{c} -2 \sqrt {2}\, \cos \left (\frac {\sqrt {2}\, x}{2}\right ) \\ 2 \sin \left (\frac {\sqrt {2}\, x}{2}\right ) \\ \sqrt {2}\, \cos \left (\frac {\sqrt {2}\, x}{2}\right ) \\ -\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {General solution to the system of ODEs}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}=\mathit {C1} {\moverset {\rightarrow }{y}}_{1}\left (x \right )+\mathit {C2} {\moverset {\rightarrow }{y}}_{2}\left (x \right )+\mathit {C3} {\moverset {\rightarrow }{y}}_{3}\left (x \right )+\mathit {C4} {\moverset {\rightarrow }{y}}_{4}\left (x \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}=\left [\begin {array}{c} -2 \mathit {C4} \sqrt {2}\, \cos \left (\frac {\sqrt {2}\, x}{2}\right )-2 \mathit {C3} \sqrt {2}\, \sin \left (\frac {\sqrt {2}\, x}{2}\right )-\frac {3 \mathit {C2} \sqrt {3}\, \cos \left (\frac {2 \sqrt {3}\, x}{3}\right )}{8}-\frac {3 \mathit {C1} \sqrt {3}\, \sin \left (\frac {2 \sqrt {3}\, x}{3}\right )}{8} \\ 2 \mathit {C4} \sin \left (\frac {\sqrt {2}\, x}{2}\right )-2 \mathit {C3} \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\frac {3 \mathit {C2} \sin \left (\frac {2 \sqrt {3}\, x}{3}\right )}{4}-\frac {3 \mathit {C1} \cos \left (\frac {2 \sqrt {3}\, x}{3}\right )}{4} \\ \mathit {C4} \sqrt {2}\, \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\mathit {C3} \sqrt {2}\, \sin \left (\frac {\sqrt {2}\, x}{2}\right )+\frac {\mathit {C2} \sqrt {3}\, \cos \left (\frac {2 \sqrt {3}\, x}{3}\right )}{2}+\frac {\mathit {C1} \sqrt {3}\, \sin \left (\frac {2 \sqrt {3}\, x}{3}\right )}{2} \\ -\mathit {C4} \sin \left (\frac {\sqrt {2}\, x}{2}\right )+\mathit {C3} \cos \left (\frac {\sqrt {2}\, x}{2}\right )-\mathit {C2} \sin \left (\frac {2 \sqrt {3}\, x}{3}\right )+\mathit {C1} \cos \left (\frac {2 \sqrt {3}\, x}{3}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y \left (x \right )=-2 \mathit {C4} \sqrt {2}\, \cos \left (\frac {\sqrt {2}\, x}{2}\right )-2 \mathit {C3} \sqrt {2}\, \sin \left (\frac {\sqrt {2}\, x}{2}\right )-\frac {3 \mathit {C2} \sqrt {3}\, \cos \left (\frac {2 \sqrt {3}\, x}{3}\right )}{8}-\frac {3 \mathit {C1} \sqrt {3}\, \sin \left (\frac {2 \sqrt {3}\, x}{3}\right )}{8} \end {array} \]

2.8.3 Maple trace

`Methods for high order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
<- constant coefficients successful`
 

2.8.4 Maple dsolve solution

Solving time : 0.002 (sec)
Leaf size : 41

dsolve(6*diff(diff(diff(diff(y(x),x),x),x),x)+11*diff(diff(y(x),x),x)+4*y(x) = 0, 
       y(x),singsol=all)
 

\[ y = c_1 \sin \left (\frac {2 \sqrt {3}\, x}{3}\right )+c_2 \cos \left (\frac {2 \sqrt {3}\, x}{3}\right )+c_3 \sin \left (\frac {\sqrt {2}\, x}{2}\right )+c_4 \cos \left (\frac {\sqrt {2}\, x}{2}\right ) \]

2.8.5 Mathematica DSolve solution

Solving time : 0.007 (sec)
Leaf size : 94

DSolve[{D[y[x],{x,4}]+11*D[y[x],{x,2}]+4*y[x]==0,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_3 \cos \left (\sqrt {\frac {1}{2} \left (11-\sqrt {105}\right )} x\right )+c_1 \cos \left (\sqrt {\frac {1}{2} \left (11+\sqrt {105}\right )} x\right )+c_4 \sin \left (\sqrt {\frac {1}{2} \left (11-\sqrt {105}\right )} x\right )+c_2 \sin \left (\sqrt {\frac {1}{2} \left (11+\sqrt {105}\right )} x\right ) \]