Internal problem ID [4592]
Internal file name [OUTPUT/4085_Sunday_June_05_2022_12_20_19_PM_87601993/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 20.22, 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]]
\[ \boxed {y^{\prime \prime \prime \prime }-8 y^{\prime \prime }+36 y=0} \] The characteristic equation is \[ \lambda ^{4}-8 \lambda ^{2}+36 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= \sqrt {5}-i\\ \lambda _2 &= i-\sqrt {5}\\ \lambda _3 &= \sqrt {5}+i\\ \lambda _4 &= -i-\sqrt {5} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (\sqrt {5}+i\right ) x} c_{1} +{\mathrm e}^{\left (-i-\sqrt {5}\right ) x} c_{2} +{\mathrm e}^{\left (i-\sqrt {5}\right ) x} c_{3} +{\mathrm e}^{\left (\sqrt {5}-i\right ) x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{\left (\sqrt {5}+i\right ) x}\\ y_2 &= {\mathrm e}^{\left (-i-\sqrt {5}\right ) x}\\ y_3 &= {\mathrm e}^{\left (i-\sqrt {5}\right ) x}\\ y_4 &= {\mathrm e}^{\left (\sqrt {5}-i\right ) x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\left (\sqrt {5}+i\right ) x} c_{1} +{\mathrm e}^{\left (-i-\sqrt {5}\right ) x} c_{2} +{\mathrm e}^{\left (i-\sqrt {5}\right ) x} c_{3} +{\mathrm e}^{\left (\sqrt {5}-i\right ) x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{\left (\sqrt {5}+i\right ) x} c_{1} +{\mathrm e}^{\left (-i-\sqrt {5}\right ) x} c_{2} +{\mathrm e}^{\left (i-\sqrt {5}\right ) x} c_{3} +{\mathrm e}^{\left (\sqrt {5}-i\right ) x} c_{4} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime \prime }-8 y^{\prime \prime }+36 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 )=8 y_{3}\left (x \right )-36 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 )=8 y_{3}\left (x \right )-36 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 \\ -36 & 0 & 8 & 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 \\ -36 & 0 & 8 & 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 {5}, \left [\begin {array}{c} \frac {1}{\left (-\mathrm {I}-\sqrt {5}\right )^{3}} \\ \frac {1}{\left (-\mathrm {I}-\sqrt {5}\right )^{2}} \\ \frac {1}{-\mathrm {I}-\sqrt {5}} \\ 1 \end {array}\right ]\right ], \left [\mathrm {I}-\sqrt {5}, \left [\begin {array}{c} \frac {1}{\left (\mathrm {I}-\sqrt {5}\right )^{3}} \\ \frac {1}{\left (\mathrm {I}-\sqrt {5}\right )^{2}} \\ \frac {1}{\mathrm {I}-\sqrt {5}} \\ 1 \end {array}\right ]\right ], \left [\sqrt {5}-\mathrm {I}, \left [\begin {array}{c} \frac {1}{\left (\sqrt {5}-\mathrm {I}\right )^{3}} \\ \frac {1}{\left (\sqrt {5}-\mathrm {I}\right )^{2}} \\ \frac {1}{\sqrt {5}-\mathrm {I}} \\ 1 \end {array}\right ]\right ], \left [\sqrt {5}+\mathrm {I}, \left [\begin {array}{c} \frac {1}{\left (\sqrt {5}+\mathrm {I}\right )^{3}} \\ \frac {1}{\left (\sqrt {5}+\mathrm {I}\right )^{2}} \\ \frac {1}{\sqrt {5}+\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 {5}, \left [\begin {array}{c} \frac {1}{\left (-\mathrm {I}-\sqrt {5}\right )^{3}} \\ \frac {1}{\left (-\mathrm {I}-\sqrt {5}\right )^{2}} \\ \frac {1}{-\mathrm {I}-\sqrt {5}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-\mathrm {I}-\sqrt {5}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-\mathrm {I}-\sqrt {5}\right )^{3}} \\ \frac {1}{\left (-\mathrm {I}-\sqrt {5}\right )^{2}} \\ \frac {1}{-\mathrm {I}-\sqrt {5}} \\ 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 {5}\, x}\cdot \left (-\mathrm {I} \sin \left (x \right )+\cos \left (x \right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (-\mathrm {I}-\sqrt {5}\right )^{3}} \\ \frac {1}{\left (-\mathrm {I}-\sqrt {5}\right )^{2}} \\ \frac {1}{-\mathrm {I}-\sqrt {5}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-\sqrt {5}\, x}\cdot \left [\begin {array}{c} \frac {-\mathrm {I} \sin \left (x \right )+\cos \left (x \right )}{\left (-\mathrm {I}-\sqrt {5}\right )^{3}} \\ \frac {-\mathrm {I} \sin \left (x \right )+\cos \left (x \right )}{\left (-\mathrm {I}-\sqrt {5}\right )^{2}} \\ \frac {-\mathrm {I} \sin \left (x \right )+\cos \left (x \right )}{-\mathrm {I}-\sqrt {5}} \\ -\mathrm {I} \sin \left (x \right )+\cos \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 {5}\, x}\cdot \left [\begin {array}{c} -\frac {\cos \left (x \right ) \sqrt {5}}{108}+\frac {7 \sin \left (x \right )}{108} \\ \frac {\cos \left (x \right )}{9}-\frac {\sin \left (x \right ) \sqrt {5}}{18} \\ -\frac {\cos \left (x \right ) \sqrt {5}}{6}+\frac {\sin \left (x \right )}{6} \\ \cos \left (x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{2}\left (x \right )={\mathrm e}^{-\sqrt {5}\, x}\cdot \left [\begin {array}{c} \frac {7 \cos \left (x \right )}{108}+\frac {\sin \left (x \right ) \sqrt {5}}{108} \\ -\frac {\cos \left (x \right ) \sqrt {5}}{18}-\frac {\sin \left (x \right )}{9} \\ \frac {\cos \left (x \right )}{6}+\frac {\sin \left (x \right ) \sqrt {5}}{6} \\ -\sin \left (x \right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [\sqrt {5}-\mathrm {I}, \left [\begin {array}{c} \frac {1}{\left (\sqrt {5}-\mathrm {I}\right )^{3}} \\ \frac {1}{\left (\sqrt {5}-\mathrm {I}\right )^{2}} \\ \frac {1}{\sqrt {5}-\mathrm {I}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\sqrt {5}-\mathrm {I}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (\sqrt {5}-\mathrm {I}\right )^{3}} \\ \frac {1}{\left (\sqrt {5}-\mathrm {I}\right )^{2}} \\ \frac {1}{\sqrt {5}-\mathrm {I}} \\ 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 {5}\, x}\cdot \left (-\mathrm {I} \sin \left (x \right )+\cos \left (x \right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (\sqrt {5}-\mathrm {I}\right )^{3}} \\ \frac {1}{\left (\sqrt {5}-\mathrm {I}\right )^{2}} \\ \frac {1}{\sqrt {5}-\mathrm {I}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\sqrt {5}\, x}\cdot \left [\begin {array}{c} \frac {-\mathrm {I} \sin \left (x \right )+\cos \left (x \right )}{\left (\sqrt {5}-\mathrm {I}\right )^{3}} \\ \frac {-\mathrm {I} \sin \left (x \right )+\cos \left (x \right )}{\left (\sqrt {5}-\mathrm {I}\right )^{2}} \\ \frac {-\mathrm {I} \sin \left (x \right )+\cos \left (x \right )}{\sqrt {5}-\mathrm {I}} \\ -\mathrm {I} \sin \left (x \right )+\cos \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 {5}\, x}\cdot \left [\begin {array}{c} \frac {\cos \left (x \right ) \sqrt {5}}{108}+\frac {7 \sin \left (x \right )}{108} \\ \frac {\cos \left (x \right )}{9}+\frac {\sin \left (x \right ) \sqrt {5}}{18} \\ \frac {\cos \left (x \right ) \sqrt {5}}{6}+\frac {\sin \left (x \right )}{6} \\ \cos \left (x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{4}\left (x \right )={\mathrm e}^{\sqrt {5}\, x}\cdot \left [\begin {array}{c} \frac {7 \cos \left (x \right )}{108}-\frac {\sin \left (x \right ) \sqrt {5}}{108} \\ \frac {\cos \left (x \right ) \sqrt {5}}{18}-\frac {\sin \left (x \right )}{9} \\ \frac {\cos \left (x \right )}{6}-\frac {\sin \left (x \right ) \sqrt {5}}{6} \\ -\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 {5}\, x}\cdot \left [\begin {array}{c} -\frac {\cos \left (x \right ) \sqrt {5}}{108}+\frac {7 \sin \left (x \right )}{108} \\ \frac {\cos \left (x \right )}{9}-\frac {\sin \left (x \right ) \sqrt {5}}{18} \\ -\frac {\cos \left (x \right ) \sqrt {5}}{6}+\frac {\sin \left (x \right )}{6} \\ \cos \left (x \right ) \end {array}\right ]+c_{2} {\mathrm e}^{-\sqrt {5}\, x}\cdot \left [\begin {array}{c} \frac {7 \cos \left (x \right )}{108}+\frac {\sin \left (x \right ) \sqrt {5}}{108} \\ -\frac {\cos \left (x \right ) \sqrt {5}}{18}-\frac {\sin \left (x \right )}{9} \\ \frac {\cos \left (x \right )}{6}+\frac {\sin \left (x \right ) \sqrt {5}}{6} \\ -\sin \left (x \right ) \end {array}\right ]+c_{3} {\mathrm e}^{\sqrt {5}\, x}\cdot \left [\begin {array}{c} \frac {\cos \left (x \right ) \sqrt {5}}{108}+\frac {7 \sin \left (x \right )}{108} \\ \frac {\cos \left (x \right )}{9}+\frac {\sin \left (x \right ) \sqrt {5}}{18} \\ \frac {\cos \left (x \right ) \sqrt {5}}{6}+\frac {\sin \left (x \right )}{6} \\ \cos \left (x \right ) \end {array}\right ]+c_{4} {\mathrm e}^{\sqrt {5}\, x}\cdot \left [\begin {array}{c} \frac {7 \cos \left (x \right )}{108}-\frac {\sin \left (x \right ) \sqrt {5}}{108} \\ \frac {\cos \left (x \right ) \sqrt {5}}{18}-\frac {\sin \left (x \right )}{9} \\ \frac {\cos \left (x \right )}{6}-\frac {\sin \left (x \right ) \sqrt {5}}{6} \\ -\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 (\left (-\cos \left (x \right ) c_{1} +c_{2} \sin \left (x \right )\right ) \sqrt {5}+7 \sin \left (x \right ) c_{1} +7 c_{2} \cos \left (x \right )\right ) {\mathrm e}^{-\sqrt {5}\, x}}{108}+\frac {{\mathrm e}^{\sqrt {5}\, x} \left (\left (\cos \left (x \right ) c_{3} -\sin \left (x \right ) c_{4} \right ) \sqrt {5}+7 \sin \left (x \right ) c_{3} +7 c_{4} \cos \left (x \right )\right )}{108} \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: 0.016 (sec). Leaf size: 48
dsolve(diff(y(x),x$4)-8*diff(y(x),x$2)+36*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{\sqrt {5}\, x} \sin \left (x \right )-c_{2} {\mathrm e}^{-\sqrt {5}\, x} \sin \left (x \right )+c_{3} {\mathrm e}^{\sqrt {5}\, x} \cos \left (x \right )+c_{4} {\mathrm e}^{-\sqrt {5}\, x} \cos \left (x \right ) \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 142
DSolve[y''''[x]-8*y''[x]+36*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\sqrt {6} x \cos \left (\frac {1}{2} \arctan \left (\frac {\sqrt {5}}{2}\right )\right )} \left (\left (c_3 e^{2 \sqrt {6} x \cos \left (\frac {1}{2} \arctan \left (\frac {\sqrt {5}}{2}\right )\right )}+c_2\right ) \cos \left (\sqrt {6} x \sin \left (\frac {1}{2} \arctan \left (\frac {\sqrt {5}}{2}\right )\right )\right )+\sin \left (\sqrt {6} x \sin \left (\frac {1}{2} \arctan \left (\frac {\sqrt {5}}{2}\right )\right )\right ) \left (c_1 e^{2 \sqrt {6} x \cos \left (\frac {1}{2} \arctan \left (\frac {\sqrt {5}}{2}\right )\right )}+c_4\right )\right ) \]