10.16 problem 16

Internal problem ID [11745]
Internal file name [OUTPUT/11755_Thursday_April_11_2024_08_49_24_PM_2170690/index.tex]

Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section: Chapter 4, Section 4.2. The homogeneous linear equation with constant coefficients. Exercises page 135
Problem number: 16.
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]]

\[ \boxed {y^{\prime \prime \prime }+4 y^{\prime \prime }+5 y^{\prime }+6 y=0} \] The characteristic equation is \[ \lambda ^{3}+4 \lambda ^{2}+5 \lambda +6 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -3\\ \lambda _2 &= -\frac {1}{2}-\frac {i \sqrt {7}}{2}\\ \lambda _3 &= -\frac {1}{2}+\frac {i \sqrt {7}}{2} \end {align*}

Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{-3 x} c_{1} +{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {7}}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {7}}{2}\right ) x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-3 x}\\ y_2 &= {\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {7}}{2}\right ) x}\\ y_3 &= {\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {7}}{2}\right ) x} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-3 x} c_{1} +{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {7}}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {7}}{2}\right ) x} c_{3} \\ \end{align*}

Verification of solutions

\[ y = {\mathrm e}^{-3 x} c_{1} +{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {7}}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {7}}{2}\right ) x} c_{3} \] Verified OK.

10.16.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }+4 \frac {d}{d x}y^{\prime }+5 y^{\prime }+6 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 )=-4 y_{3}\left (x \right )-5 y_{2}\left (x \right )-6 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 )=-4 y_{3}\left (x \right )-5 y_{2}\left (x \right )-6 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 \\ -6 & -5 & -4 \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 \\ -6 & -5 & -4 \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 [-\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}\right )^{2}} \\ \frac {1}{-\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}} \\ 1 \end {array}\right ]\right ], \left [-\frac {1}{2}+\frac {\mathrm {I} \sqrt {7}}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {7}}{2}\right )^{2}} \\ \frac {1}{-\frac {1}{2}+\frac {\mathrm {I} \sqrt {7}}{2}} \\ 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 [-\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}\right )^{2}} \\ \frac {1}{-\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}\right )^{2}} \\ \frac {1}{-\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{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 \\ {} & {} & {\mathrm e}^{-\frac {x}{2}}\cdot \left (\cos \left (\frac {\sqrt {7}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {7}\, x}{2}\right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}\right )^{2}} \\ \frac {1}{-\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-\frac {x}{2}}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {7}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {7}\, x}{2}\right )}{\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}\right )^{2}} \\ \frac {\cos \left (\frac {\sqrt {7}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {7}\, x}{2}\right )}{-\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}} \\ \cos \left (\frac {\sqrt {7}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {7}\, 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}}_{2}\left (x \right )={\mathrm e}^{-\frac {x}{2}}\cdot \left [\begin {array}{c} -\frac {3 \cos \left (\frac {\sqrt {7}\, x}{2}\right )}{8}-\frac {\sin \left (\frac {\sqrt {7}\, x}{2}\right ) \sqrt {7}}{8} \\ -\frac {\cos \left (\frac {\sqrt {7}\, x}{2}\right )}{4}+\frac {\sin \left (\frac {\sqrt {7}\, x}{2}\right ) \sqrt {7}}{4} \\ \cos \left (\frac {\sqrt {7}\, x}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{3}\left (x \right )={\mathrm e}^{-\frac {x}{2}}\cdot \left [\begin {array}{c} -\frac {\cos \left (\frac {\sqrt {7}\, x}{2}\right ) \sqrt {7}}{8}+\frac {3 \sin \left (\frac {\sqrt {7}\, x}{2}\right )}{8} \\ \frac {\cos \left (\frac {\sqrt {7}\, x}{2}\right ) \sqrt {7}}{4}+\frac {\sin \left (\frac {\sqrt {7}\, x}{2}\right )}{4} \\ -\sin \left (\frac {\sqrt {7}\, x}{2}\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}^{-3 x} c_{1} \cdot \left [\begin {array}{c} \frac {1}{9} \\ -\frac {1}{3} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{-\frac {x}{2}}\cdot \left [\begin {array}{c} -\frac {3 \cos \left (\frac {\sqrt {7}\, x}{2}\right )}{8}-\frac {\sin \left (\frac {\sqrt {7}\, x}{2}\right ) \sqrt {7}}{8} \\ -\frac {\cos \left (\frac {\sqrt {7}\, x}{2}\right )}{4}+\frac {\sin \left (\frac {\sqrt {7}\, x}{2}\right ) \sqrt {7}}{4} \\ \cos \left (\frac {\sqrt {7}\, x}{2}\right ) \end {array}\right ]+c_{3} {\mathrm e}^{-\frac {x}{2}}\cdot \left [\begin {array}{c} -\frac {\cos \left (\frac {\sqrt {7}\, x}{2}\right ) \sqrt {7}}{8}+\frac {3 \sin \left (\frac {\sqrt {7}\, x}{2}\right )}{8} \\ \frac {\cos \left (\frac {\sqrt {7}\, x}{2}\right ) \sqrt {7}}{4}+\frac {\sin \left (\frac {\sqrt {7}\, x}{2}\right )}{4} \\ -\sin \left (\frac {\sqrt {7}\, x}{2}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=-\frac {3 \,{\mathrm e}^{-\frac {x}{2}} \left (\frac {c_{3} \sqrt {7}}{3}+c_{2} \right ) \cos \left (\frac {\sqrt {7}\, x}{2}\right )}{8}-\frac {{\mathrm e}^{-\frac {x}{2}} \left (c_{2} \sqrt {7}-3 c_{3} \right ) \sin \left (\frac {\sqrt {7}\, x}{2}\right )}{8}+\frac {{\mathrm e}^{-3 x} c_{1}}{9} \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: 37

dsolve(diff(y(x),x$3)+4*diff(y(x),x$2)+5*diff(y(x),x)+6*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = c_{1} {\mathrm e}^{-3 x}+c_{2} {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {7}\, x}{2}\right )+c_{3} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {7}\, x}{2}\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 56

DSolve[y'''[x]+4*y''[x]+5*y'[x]+6*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-3 x} \left (c_2 e^{5 x/2} \cos \left (\frac {\sqrt {7} x}{2}\right )+c_1 e^{5 x/2} \sin \left (\frac {\sqrt {7} x}{2}\right )+c_3\right ) \]