9.23 problem 37

Internal problem ID [2109]
Internal file name [OUTPUT/2109_Sunday_February_25_2024_06_49_50_AM_23139370/index.tex]

Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section: Exercise 17, page 78
Problem number: 37.
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]]

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

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

Summary

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

Verification of solutions

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

9.23.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 y^{\prime \prime \prime }+2 y^{\prime \prime }-4 y^{\prime }+y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & y^{\prime \prime \prime } \\ \bullet & {} & \textrm {Isolate 3rd derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }=-\frac {y^{\prime \prime }}{2}+y^{\prime }-\frac {y}{4} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }+\frac {y^{\prime \prime }}{2}-y^{\prime }+\frac {y}{4}=0 \\ \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 )=-\frac {y_{3}\left (x \right )}{2}+y_{2}\left (x \right )-\frac {y_{1}\left (x \right )}{4} \\ & {} & \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 )=-\frac {y_{3}\left (x \right )}{2}+y_{2}\left (x \right )-\frac {y_{1}\left (x \right )}{4}\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 \\ -\frac {1}{4} & 1 & -\frac {1}{2} \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 \\ -\frac {1}{4} & 1 & -\frac {1}{2} \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 [\frac {1}{2}, \left [\begin {array}{c} 4 \\ 2 \\ 1 \end {array}\right ]\right ], \left [-\frac {1}{2}-\frac {\sqrt {3}}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {1}{2}-\frac {\sqrt {3}}{2}} \\ 1 \end {array}\right ]\right ], \left [\frac {\sqrt {3}}{2}-\frac {1}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}} \\ \frac {1}{\frac {\sqrt {3}}{2}-\frac {1}{2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [\frac {1}{2}, \left [\begin {array}{c} 4 \\ 2 \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{\frac {x}{2}}\cdot \left [\begin {array}{c} 4 \\ 2 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {1}{2}-\frac {\sqrt {3}}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {1}{2}-\frac {\sqrt {3}}{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{2}={\mathrm e}^{\left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {1}{2}-\frac {\sqrt {3}}{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [\frac {\sqrt {3}}{2}-\frac {1}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}} \\ \frac {1}{\frac {\sqrt {3}}{2}-\frac {1}{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{3}={\mathrm e}^{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}} \\ \frac {1}{\frac {\sqrt {3}}{2}-\frac {1}{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}}=c_{1} {\mathrm e}^{\frac {x}{2}}\cdot \left [\begin {array}{c} 4 \\ 2 \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{\left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {1}{2}-\frac {\sqrt {3}}{2}} \\ 1 \end {array}\right ]+{\mathrm e}^{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right ) x} c_{3} \cdot \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}} \\ \frac {1}{\frac {\sqrt {3}}{2}-\frac {1}{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=-2 c_{2} \left (\sqrt {3}-2\right ) {\mathrm e}^{-\frac {\left (1+\sqrt {3}\right ) x}{2}}+2 c_{3} \left (2+\sqrt {3}\right ) {\mathrm e}^{\frac {\left (\sqrt {3}-1\right ) x}{2}}+4 c_{1} {\mathrm e}^{\frac {x}{2}} \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: 0.016 (sec). Leaf size: 33

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 50

DSolve[4*y'''[x]+2*y''[x]-4*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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