Internal problem ID [13625]
Internal file name [OUTPUT/12797_Saturday_February_17_2024_08_44_52_AM_56013306/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 19. Arbitrary Homogeneous linear equations with constant coefficients.
Additional exercises page 369
Problem number: 19.2 (d).
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]]
\[ \boxed {y^{\prime \prime \prime }-9 y^{\prime \prime }+31 y^{\prime }-39 y=0} \] The characteristic equation is \[ \lambda ^{3}-9 \lambda ^{2}+31 \lambda -39 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 3\\ \lambda _2 &= 3-2 i\\ \lambda _3 &= 3+2 i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (3+2 i\right ) x} c_{1} +c_{2} {\mathrm e}^{3 x}+{\mathrm e}^{\left (3-2 i\right ) x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{\left (3+2 i\right ) x}\\ y_2 &= {\mathrm e}^{3 x}\\ y_3 &= {\mathrm e}^{\left (3-2 i\right ) x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\left (3+2 i\right ) x} c_{1} +c_{2} {\mathrm e}^{3 x}+{\mathrm e}^{\left (3-2 i\right ) x} c_{3} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{\left (3+2 i\right ) x} c_{1} +c_{2} {\mathrm e}^{3 x}+{\mathrm e}^{\left (3-2 i\right ) x} c_{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }-9 y^{\prime \prime }+31 y^{\prime }-39 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & y^{\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 {Isolate for}\hspace {3pt} y_{3}^{\prime }\left (x \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{3}^{\prime }\left (x \right )=9 y_{3}\left (x \right )-31 y_{2}\left (x \right )+39 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 )=9 y_{3}\left (x \right )-31 y_{2}\left (x \right )+39 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 \\ 39 & -31 & 9 \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 \\ 39 & -31 & 9 \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 [3, \left [\begin {array}{c} \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]\right ], \left [3-2 \,\mathrm {I}, \left [\begin {array}{c} \frac {5}{169}+\frac {12 \,\mathrm {I}}{169} \\ \frac {3}{13}+\frac {2 \,\mathrm {I}}{13} \\ 1 \end {array}\right ]\right ], \left [3+2 \,\mathrm {I}, \left [\begin {array}{c} \frac {5}{169}-\frac {12 \,\mathrm {I}}{169} \\ \frac {3}{13}-\frac {2 \,\mathrm {I}}{13} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [3, \left [\begin {array}{c} \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{3 x}\cdot \left [\begin {array}{c} \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [3-2 \,\mathrm {I}, \left [\begin {array}{c} \frac {5}{169}+\frac {12 \,\mathrm {I}}{169} \\ \frac {3}{13}+\frac {2 \,\mathrm {I}}{13} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (3-2 \,\mathrm {I}\right ) x}\cdot \left [\begin {array}{c} \frac {5}{169}+\frac {12 \,\mathrm {I}}{169} \\ \frac {3}{13}+\frac {2 \,\mathrm {I}}{13} \\ 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}^{3 x}\cdot \left (\cos \left (2 x \right )-\mathrm {I} \sin \left (2 x \right )\right )\cdot \left [\begin {array}{c} \frac {5}{169}+\frac {12 \,\mathrm {I}}{169} \\ \frac {3}{13}+\frac {2 \,\mathrm {I}}{13} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{3 x}\cdot \left [\begin {array}{c} \left (\frac {5}{169}+\frac {12 \,\mathrm {I}}{169}\right ) \left (\cos \left (2 x \right )-\mathrm {I} \sin \left (2 x \right )\right ) \\ \left (\frac {3}{13}+\frac {2 \,\mathrm {I}}{13}\right ) \left (\cos \left (2 x \right )-\mathrm {I} \sin \left (2 x \right )\right ) \\ \cos \left (2 x \right )-\mathrm {I} \sin \left (2 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}^{3 x}\cdot \left [\begin {array}{c} \frac {5 \cos \left (2 x \right )}{169}+\frac {12 \sin \left (2 x \right )}{169} \\ \frac {3 \cos \left (2 x \right )}{13}+\frac {2 \sin \left (2 x \right )}{13} \\ \cos \left (2 x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{3}\left (x \right )={\mathrm e}^{3 x}\cdot \left [\begin {array}{c} -\frac {5 \sin \left (2 x \right )}{169}+\frac {12 \cos \left (2 x \right )}{169} \\ -\frac {3 \sin \left (2 x \right )}{13}+\frac {2 \cos \left (2 x \right )}{13} \\ -\sin \left (2 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}}=c_{1} {\mathrm e}^{3 x}\cdot \left [\begin {array}{c} \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{3 x}\cdot \left [\begin {array}{c} \frac {5 \cos \left (2 x \right )}{169}+\frac {12 \sin \left (2 x \right )}{169} \\ \frac {3 \cos \left (2 x \right )}{13}+\frac {2 \sin \left (2 x \right )}{13} \\ \cos \left (2 x \right ) \end {array}\right ]+c_{3} {\mathrm e}^{3 x}\cdot \left [\begin {array}{c} -\frac {5 \sin \left (2 x \right )}{169}+\frac {12 \cos \left (2 x \right )}{169} \\ -\frac {3 \sin \left (2 x \right )}{13}+\frac {2 \cos \left (2 x \right )}{13} \\ -\sin \left (2 x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {{\mathrm e}^{3 x} \left (45 c_{2} \cos \left (2 x \right )+108 c_{3} \cos \left (2 x \right )+108 \sin \left (2 x \right ) c_{2} -45 c_{3} \sin \left (2 x \right )+169 c_{1} \right )}{1521} \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: 23
dsolve(diff(y(x),x$3)-9*diff(y(x),x$2)+31*diff(y(x),x)-39*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{3 x} \left (c_{1} +c_{2} \sin \left (2 x \right )+c_{3} \cos \left (2 x \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 28
DSolve[y'''[x]-9*y''[x]+31*y'[x]-39*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{3 x} (c_2 \cos (2 x)+c_1 \sin (2 x)+c_3) \]