23.2 problem Ex 3

Internal problem ID [11251]
Internal file name [OUTPUT/10237_Sunday_December_11_2022_01_27_03_AM_84344330/index.tex]

Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section: Chapter VII, Linear differential equations with constant coefficients. Article 45. Roots of auxiliary equation complex. Page 95
Problem number: Ex 3.
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]]

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

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

Summary

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

Verification of solutions

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

23.2.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }=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 )-y_{2}\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 )-y_{2}\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 \\ 0 & -1 & 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 \\ 0 & -1 & 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 [0, \left [\begin {array}{c} 1 \\ 0 \\ 0 \end {array}\right ]\right ], \left [\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {1}{\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}} \\ 1 \end {array}\right ]\right ], \left [\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {1}{\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [0, \left [\begin {array}{c} 1 \\ 0 \\ 0 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}=\left [\begin {array}{c} 1 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {1}{\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {1}{\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{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 {3}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, x}{2}\right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {1}{\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{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 {3}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}} \\ \cos \left (\frac {\sqrt {3}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, 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 {\cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2} \\ \frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2} \\ \cos \left (\frac {\sqrt {3}\, 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 {3}\, x}{2}\right ) \sqrt {3}}{2}+\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2} \\ \frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2}-\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2} \\ -\sin \left (\frac {\sqrt {3}\, 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}^{\frac {x}{2}} c_{2} \cdot \left [\begin {array}{c} -\frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2} \\ \frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2} \\ \cos \left (\frac {\sqrt {3}\, x}{2}\right ) \end {array}\right ]+{\mathrm e}^{\frac {x}{2}} c_{3} \cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2}+\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2} \\ \frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2}-\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2} \\ -\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \end {array}\right ]+\left [\begin {array}{c} c_{1} \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=-\frac {{\mathrm e}^{\frac {x}{2}} \left (-c_{3} \sqrt {3}+c_{2} \right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {{\mathrm e}^{\frac {x}{2}} \left (c_{2} \sqrt {3}+c_{3} \right ) \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+c_{1} \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: 32

dsolve(diff(y(x),x$3)-diff(y(x),x$2)+diff(y(x),x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.799 (sec). Leaf size: 75

DSolve[y'''[x]-y''[x]+y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} \left (c_1-\sqrt {3} c_2\right ) e^{x/2} \cos \left (\frac {\sqrt {3} x}{2}\right )+\frac {1}{2} \left (\sqrt {3} c_1+c_2\right ) e^{x/2} \sin \left (\frac {\sqrt {3} x}{2}\right )+c_3 \]