Internal problem ID [4577]
Internal file name [OUTPUT/4070_Sunday_June_05_2022_12_18_28_PM_73748754/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 20.7, page 220.
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]]
\[ \boxed {y^{\prime \prime \prime }+y^{\prime \prime }-10 y^{\prime }-6 y=0} \] The characteristic equation is \[ \lambda ^{3}+\lambda ^{2}-10 \lambda -6 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 3\\ \lambda _2 &= \sqrt {2}-2\\ \lambda _3 &= -\sqrt {2}-2 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{3 x} c_{1} +{\mathrm e}^{\left (-\sqrt {2}-2\right ) x} c_{2} +{\mathrm e}^{\left (\sqrt {2}-2\right ) x} c_{3} \] 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 (-\sqrt {2}-2\right ) x}\\ y_3 &= {\mathrm e}^{\left (\sqrt {2}-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 (-\sqrt {2}-2\right ) x} c_{2} +{\mathrm e}^{\left (\sqrt {2}-2\right ) x} c_{3} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{3 x} c_{1} +{\mathrm e}^{\left (-\sqrt {2}-2\right ) x} c_{2} +{\mathrm e}^{\left (\sqrt {2}-2\right ) x} c_{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }+y^{\prime \prime }-10 y^{\prime }-6 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 )=-y_{3}\left (x \right )+10 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 )=-y_{3}\left (x \right )+10 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 & 10 & -1 \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 & 10 & -1 \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 [-\sqrt {2}-2, \left [\begin {array}{c} \frac {1}{\left (-\sqrt {2}-2\right )^{2}} \\ \frac {1}{-\sqrt {2}-2} \\ 1 \end {array}\right ]\right ], \left [\sqrt {2}-2, \left [\begin {array}{c} \frac {1}{\left (\sqrt {2}-2\right )^{2}} \\ \frac {1}{\sqrt {2}-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 eigenpair}\hspace {3pt} \\ {} & {} & \left [-\sqrt {2}-2, \left [\begin {array}{c} \frac {1}{\left (-\sqrt {2}-2\right )^{2}} \\ \frac {1}{-\sqrt {2}-2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{2}={\mathrm e}^{\left (-\sqrt {2}-2\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-\sqrt {2}-2\right )^{2}} \\ \frac {1}{-\sqrt {2}-2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [\sqrt {2}-2, \left [\begin {array}{c} \frac {1}{\left (\sqrt {2}-2\right )^{2}} \\ \frac {1}{\sqrt {2}-2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{3}={\mathrm e}^{\left (\sqrt {2}-2\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (\sqrt {2}-2\right )^{2}} \\ \frac {1}{\sqrt {2}-2} \\ 1 \end {array}\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}+c_{3} {\moverset {\rightarrow }{y}}_{3} \\ \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 ]+{\mathrm e}^{\left (-\sqrt {2}-2\right ) x} c_{2} \cdot \left [\begin {array}{c} \frac {1}{\left (-\sqrt {2}-2\right )^{2}} \\ \frac {1}{-\sqrt {2}-2} \\ 1 \end {array}\right ]+{\mathrm e}^{\left (\sqrt {2}-2\right ) x} c_{3} \cdot \left [\begin {array}{c} \frac {1}{\left (\sqrt {2}-2\right )^{2}} \\ \frac {1}{\sqrt {2}-2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {c_{2} \left (3-2 \sqrt {2}\right ) {\mathrm e}^{-\left (2+\sqrt {2}\right ) x}}{2}+\frac {c_{3} \left (2 \sqrt {2}+3\right ) {\mathrm e}^{\left (\sqrt {2}-2\right ) x}}{2}+\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: 0.015 (sec). Leaf size: 32
dsolve(diff(y(x),x$3)+diff(y(x),x$2)-10*diff(y(x),x)-6*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{3 x}+c_{2} {\mathrm e}^{\left (-2+\sqrt {2}\right ) x}+c_{3} {\mathrm e}^{-\left (2+\sqrt {2}\right ) x} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 43
DSolve[y'''[x]+y''[x]-10*y'[x]-6*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 e^{-\left (\left (2+\sqrt {2}\right ) x\right )}+c_2 e^{\left (\sqrt {2}-2\right ) x}+c_3 e^{3 x} \]