Internal problem ID [3245]
Internal file name [OUTPUT/2737_Sunday_June_05_2022_08_39_53_AM_98960955/index.tex]
Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 4. Linear Differential Equations. Page 183
Problem number: 4.
ODE order: 3.
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
[[_3rd_order, _missing_x]]
\[ \boxed {y^{\prime \prime \prime }+8 y=0} \] The characteristic equation is \[ \lambda ^{3}+8 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -2\\ \lambda _2 &= 1-i \sqrt {3}\\ \lambda _3 &= 1+i \sqrt {3} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{-2 x} c_{1} +{\mathrm e}^{\left (1-i \sqrt {3}\right ) x} c_{2} +{\mathrm e}^{\left (1+i \sqrt {3}\right ) x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-2 x}\\ y_2 &= {\mathrm e}^{\left (1-i \sqrt {3}\right ) x}\\ y_3 &= {\mathrm e}^{\left (1+i \sqrt {3}\right ) x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-2 x} c_{1} +{\mathrm e}^{\left (1-i \sqrt {3}\right ) x} c_{2} +{\mathrm e}^{\left (1+i \sqrt {3}\right ) x} c_{3} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{-2 x} c_{1} +{\mathrm e}^{\left (1-i \sqrt {3}\right ) x} c_{2} +{\mathrm e}^{\left (1+i \sqrt {3}\right ) x} c_{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }+8 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \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 {Isolate for}\hspace {3pt} y_{3}^{\prime }\left (x \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{3}^{\prime }\left (x \right )=-8 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_{3}^{\prime }\left (x \right )=-8 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 ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -8 & 0 & 0 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -8 & 0 & 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 [-2, \left [\begin {array}{c} \frac {1}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ]\right ], \left [1-\mathrm {I} \sqrt {3}, \left [\begin {array}{c} \frac {1}{\left (1-\mathrm {I} \sqrt {3}\right )^{2}} \\ \frac {1}{1-\mathrm {I} \sqrt {3}} \\ 1 \end {array}\right ]\right ], \left [1+\mathrm {I} \sqrt {3}, \left [\begin {array}{c} \frac {1}{\left (1+\mathrm {I} \sqrt {3}\right )^{2}} \\ \frac {1}{1+\mathrm {I} \sqrt {3}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-2, \left [\begin {array}{c} \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}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [1-\mathrm {I} \sqrt {3}, \left [\begin {array}{c} \frac {1}{\left (1-\mathrm {I} \sqrt {3}\right )^{2}} \\ \frac {1}{1-\mathrm {I} \sqrt {3}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (1-\mathrm {I} \sqrt {3}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (1-\mathrm {I} \sqrt {3}\right )^{2}} \\ \frac {1}{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}^{x}\cdot \left (\cos \left (\sqrt {3}\, x \right )-\mathrm {I} \sin \left (\sqrt {3}\, x \right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (1-\mathrm {I} \sqrt {3}\right )^{2}} \\ \frac {1}{1-\mathrm {I} \sqrt {3}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{x}\cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {3}\, x \right )-\mathrm {I} \sin \left (\sqrt {3}\, x \right )}{\left (1-\mathrm {I} \sqrt {3}\right )^{2}} \\ \frac {\cos \left (\sqrt {3}\, x \right )-\mathrm {I} \sin \left (\sqrt {3}\, x \right )}{1-\mathrm {I} \sqrt {3}} \\ \cos \left (\sqrt {3}\, x \right )-\mathrm {I} \sin \left (\sqrt {3}\, x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{2}\left (x \right )={\mathrm e}^{x}\cdot \left [\begin {array}{c} -\frac {\cos \left (\sqrt {3}\, x \right )}{8}+\frac {\sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{8} \\ \frac {\cos \left (\sqrt {3}\, x \right )}{4}+\frac {\sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{4} \\ \cos \left (\sqrt {3}\, x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{3}\left (x \right )={\mathrm e}^{x}\cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {3}\, x \right ) \sqrt {3}}{8}+\frac {\sin \left (\sqrt {3}\, x \right )}{8} \\ \frac {\cos \left (\sqrt {3}\, x \right ) \sqrt {3}}{4}-\frac {\sin \left (\sqrt {3}\, x \right )}{4} \\ -\sin \left (\sqrt {3}\, 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}+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (x \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (x \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}={\mathrm e}^{-2 x} c_{1} \cdot \left [\begin {array}{c} \frac {1}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{x}\cdot \left [\begin {array}{c} -\frac {\cos \left (\sqrt {3}\, x \right )}{8}+\frac {\sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{8} \\ \frac {\cos \left (\sqrt {3}\, x \right )}{4}+\frac {\sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{4} \\ \cos \left (\sqrt {3}\, x \right ) \end {array}\right ]+{\mathrm e}^{x} c_{3} \cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {3}\, x \right ) \sqrt {3}}{8}+\frac {\sin \left (\sqrt {3}\, x \right )}{8} \\ \frac {\cos \left (\sqrt {3}\, x \right ) \sqrt {3}}{4}-\frac {\sin \left (\sqrt {3}\, x \right )}{4} \\ -\sin \left (\sqrt {3}\, x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {\left (-\frac {{\mathrm e}^{3 x} \left (-c_{3} \sqrt {3}+c_{2} \right ) \cos \left (\sqrt {3}\, x \right )}{2}+\frac {{\mathrm e}^{3 x} \left (c_{2} \sqrt {3}+c_{3} \right ) \sin \left (\sqrt {3}\, x \right )}{2}+c_{1} \right ) {\mathrm e}^{-2 x}}{4} \end {array} \]
Maple trace
`Methods for third 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: 35
dsolve(diff(y(x),x$3)+8*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{2} {\mathrm e}^{3 x} \sin \left (\sqrt {3}\, x \right )+c_{3} {\mathrm e}^{3 x} \cos \left (\sqrt {3}\, x \right )+c_{1} \right ) {\mathrm e}^{-2 x} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 42
DSolve[y'''[x]+8*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 e^{-2 x}+c_3 e^x \cos \left (\sqrt {3} x\right )+c_2 e^x \sin \left (\sqrt {3} x\right ) \]