Internal problem ID [3249]
Internal file name [OUTPUT/2741_Sunday_June_05_2022_08_39_55_AM_42962800/index.tex]
Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 4. Linear Differential Equations. Page 183
Problem number: 8.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }-4 y^{\prime \prime }+16 y=0} \] The characteristic equation is \[ \lambda ^{4}-4 \lambda ^{2}+16 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -i+\sqrt {3}\\ \lambda _2 &= i-\sqrt {3}\\ \lambda _3 &= \sqrt {3}+i\\ \lambda _4 &= -i-\sqrt {3} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (-i+\sqrt {3}\right ) x} c_{1} +{\mathrm e}^{\left (i-\sqrt {3}\right ) x} c_{2} +{\mathrm e}^{\left (\sqrt {3}+i\right ) x} c_{3} +{\mathrm e}^{\left (-i-\sqrt {3}\right ) x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{\left (-i+\sqrt {3}\right ) x}\\ y_2 &= {\mathrm e}^{\left (i-\sqrt {3}\right ) x}\\ y_3 &= {\mathrm e}^{\left (\sqrt {3}+i\right ) x}\\ y_4 &= {\mathrm e}^{\left (-i-\sqrt {3}\right ) x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\left (-i+\sqrt {3}\right ) x} c_{1} +{\mathrm e}^{\left (i-\sqrt {3}\right ) x} c_{2} +{\mathrm e}^{\left (\sqrt {3}+i\right ) x} c_{3} +{\mathrm e}^{\left (-i-\sqrt {3}\right ) x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{\left (-i+\sqrt {3}\right ) x} c_{1} +{\mathrm e}^{\left (i-\sqrt {3}\right ) x} c_{2} +{\mathrm e}^{\left (\sqrt {3}+i\right ) x} c_{3} +{\mathrm e}^{\left (-i-\sqrt {3}\right ) x} c_{4} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime }-4 \frac {d}{d x}y^{\prime }+16 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\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 )=\frac {d}{d x}y^{\prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{4}\left (x \right ) \\ {} & {} & y_{4}\left (x \right )=\frac {d}{d x}y^{\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_{3}\left (x \right )-16 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_{3}\left (x \right )-16 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 \\ -16 & 0 & 4 & 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 \\ -16 & 0 & 4 & 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 [-\mathrm {I}-\sqrt {3}, \left [\begin {array}{c} \frac {1}{\left (-\mathrm {I}-\sqrt {3}\right )^{3}} \\ \frac {1}{\left (-\mathrm {I}-\sqrt {3}\right )^{2}} \\ \frac {1}{-\mathrm {I}-\sqrt {3}} \\ 1 \end {array}\right ]\right ], \left [-\mathrm {I}+\sqrt {3}, \left [\begin {array}{c} \frac {1}{\left (-\mathrm {I}+\sqrt {3}\right )^{3}} \\ \frac {1}{\left (-\mathrm {I}+\sqrt {3}\right )^{2}} \\ \frac {1}{-\mathrm {I}+\sqrt {3}} \\ 1 \end {array}\right ]\right ], \left [\mathrm {I}-\sqrt {3}, \left [\begin {array}{c} \frac {1}{\left (\mathrm {I}-\sqrt {3}\right )^{3}} \\ \frac {1}{\left (\mathrm {I}-\sqrt {3}\right )^{2}} \\ \frac {1}{\mathrm {I}-\sqrt {3}} \\ 1 \end {array}\right ]\right ], \left [\sqrt {3}+\mathrm {I}, \left [\begin {array}{c} \frac {1}{\left (\sqrt {3}+\mathrm {I}\right )^{3}} \\ \frac {1}{\left (\sqrt {3}+\mathrm {I}\right )^{2}} \\ \frac {1}{\sqrt {3}+\mathrm {I}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-\mathrm {I}-\sqrt {3}, \left [\begin {array}{c} \frac {1}{\left (-\mathrm {I}-\sqrt {3}\right )^{3}} \\ \frac {1}{\left (-\mathrm {I}-\sqrt {3}\right )^{2}} \\ \frac {1}{-\mathrm {I}-\sqrt {3}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-\mathrm {I}-\sqrt {3}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-\mathrm {I}-\sqrt {3}\right )^{3}} \\ \frac {1}{\left (-\mathrm {I}-\sqrt {3}\right )^{2}} \\ \frac {1}{-\mathrm {I}-\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 \\ {} & {} & {\mathrm e}^{-\sqrt {3}\, x}\cdot \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (-\mathrm {I}-\sqrt {3}\right )^{3}} \\ \frac {1}{\left (-\mathrm {I}-\sqrt {3}\right )^{2}} \\ \frac {1}{-\mathrm {I}-\sqrt {3}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-\sqrt {3}\, x}\cdot \left [\begin {array}{c} \frac {\cos \left (x \right )-\mathrm {I} \sin \left (x \right )}{\left (-\mathrm {I}-\sqrt {3}\right )^{3}} \\ \frac {\cos \left (x \right )-\mathrm {I} \sin \left (x \right )}{\left (-\mathrm {I}-\sqrt {3}\right )^{2}} \\ \frac {\cos \left (x \right )-\mathrm {I} \sin \left (x \right )}{-\mathrm {I}-\sqrt {3}} \\ \cos \left (x \right )-\mathrm {I} \sin \left (x \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 )={\mathrm e}^{-\sqrt {3}\, x}\cdot \left [\begin {array}{c} \frac {\sin \left (x \right )}{8} \\ \frac {\cos \left (x \right )}{8}-\frac {\sin \left (x \right ) \sqrt {3}}{8} \\ -\frac {\cos \left (x \right ) \sqrt {3}}{4}+\frac {\sin \left (x \right )}{4} \\ \cos \left (x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{2}\left (x \right )={\mathrm e}^{-\sqrt {3}\, x}\cdot \left [\begin {array}{c} \frac {\cos \left (x \right )}{8} \\ -\frac {\cos \left (x \right ) \sqrt {3}}{8}-\frac {\sin \left (x \right )}{8} \\ \frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right ) \sqrt {3}}{4} \\ -\sin \left (x \right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-\mathrm {I}+\sqrt {3}, \left [\begin {array}{c} \frac {1}{\left (-\mathrm {I}+\sqrt {3}\right )^{3}} \\ \frac {1}{\left (-\mathrm {I}+\sqrt {3}\right )^{2}} \\ \frac {1}{-\mathrm {I}+\sqrt {3}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-\mathrm {I}+\sqrt {3}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-\mathrm {I}+\sqrt {3}\right )^{3}} \\ \frac {1}{\left (-\mathrm {I}+\sqrt {3}\right )^{2}} \\ \frac {1}{-\mathrm {I}+\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 \\ {} & {} & {\mathrm e}^{\sqrt {3}\, x}\cdot \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (-\mathrm {I}+\sqrt {3}\right )^{3}} \\ \frac {1}{\left (-\mathrm {I}+\sqrt {3}\right )^{2}} \\ \frac {1}{-\mathrm {I}+\sqrt {3}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\sqrt {3}\, x}\cdot \left [\begin {array}{c} \frac {\cos \left (x \right )-\mathrm {I} \sin \left (x \right )}{\left (-\mathrm {I}+\sqrt {3}\right )^{3}} \\ \frac {\cos \left (x \right )-\mathrm {I} \sin \left (x \right )}{\left (-\mathrm {I}+\sqrt {3}\right )^{2}} \\ \frac {\cos \left (x \right )-\mathrm {I} \sin \left (x \right )}{-\mathrm {I}+\sqrt {3}} \\ \cos \left (x \right )-\mathrm {I} \sin \left (x \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}^{\sqrt {3}\, x}\cdot \left [\begin {array}{c} \frac {\sin \left (x \right )}{8} \\ \frac {\cos \left (x \right )}{8}+\frac {\sin \left (x \right ) \sqrt {3}}{8} \\ \frac {\cos \left (x \right ) \sqrt {3}}{4}+\frac {\sin \left (x \right )}{4} \\ \cos \left (x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{4}\left (x \right )={\mathrm e}^{\sqrt {3}\, x}\cdot \left [\begin {array}{c} \frac {\cos \left (x \right )}{8} \\ \frac {\cos \left (x \right ) \sqrt {3}}{8}-\frac {\sin \left (x \right )}{8} \\ \frac {\cos \left (x \right )}{4}-\frac {\sin \left (x \right ) \sqrt {3}}{4} \\ -\sin \left (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}\left (x \right )+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (x \right )+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}}=c_{1} {\mathrm e}^{-\sqrt {3}\, x}\cdot \left [\begin {array}{c} \frac {\sin \left (x \right )}{8} \\ \frac {\cos \left (x \right )}{8}-\frac {\sin \left (x \right ) \sqrt {3}}{8} \\ -\frac {\cos \left (x \right ) \sqrt {3}}{4}+\frac {\sin \left (x \right )}{4} \\ \cos \left (x \right ) \end {array}\right ]+c_{2} {\mathrm e}^{-\sqrt {3}\, x}\cdot \left [\begin {array}{c} \frac {\cos \left (x \right )}{8} \\ -\frac {\cos \left (x \right ) \sqrt {3}}{8}-\frac {\sin \left (x \right )}{8} \\ \frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right ) \sqrt {3}}{4} \\ -\sin \left (x \right ) \end {array}\right ]+c_{3} {\mathrm e}^{\sqrt {3}\, x}\cdot \left [\begin {array}{c} \frac {\sin \left (x \right )}{8} \\ \frac {\cos \left (x \right )}{8}+\frac {\sin \left (x \right ) \sqrt {3}}{8} \\ \frac {\cos \left (x \right ) \sqrt {3}}{4}+\frac {\sin \left (x \right )}{4} \\ \cos \left (x \right ) \end {array}\right ]+c_{4} {\mathrm e}^{\sqrt {3}\, x}\cdot \left [\begin {array}{c} \frac {\cos \left (x \right )}{8} \\ \frac {\cos \left (x \right ) \sqrt {3}}{8}-\frac {\sin \left (x \right )}{8} \\ \frac {\cos \left (x \right )}{4}-\frac {\sin \left (x \right ) \sqrt {3}}{4} \\ -\sin \left (x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {\left (\sin \left (x \right ) c_{1} +\cos \left (x \right ) c_{2} \right ) {\mathrm e}^{-\sqrt {3}\, x}}{8}+\frac {{\mathrm e}^{\sqrt {3}\, x} \left (\sin \left (x \right ) c_{3} +\cos \left (x \right ) c_{4} \right )}{8} \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: 48
dsolve(diff(y(x),x$4)-4*diff(y(x),x$2)+16*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = -c_{1} {\mathrm e}^{\sqrt {3}\, x} \sin \left (x \right )+c_{2} {\mathrm e}^{-\sqrt {3}\, x} \sin \left (x \right )+c_{3} {\mathrm e}^{\sqrt {3}\, x} \cos \left (x \right )+c_{4} {\mathrm e}^{-\sqrt {3}\, x} \cos \left (x \right ) \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 55
DSolve[y''''[x]-4*y''[x]+16*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\sqrt {3} x} \left (\left (c_3 e^{2 \sqrt {3} x}+c_2\right ) \cos (x)+\left (c_1 e^{2 \sqrt {3} x}+c_4\right ) \sin (x)\right ) \]