9.21 problem 35

Internal problem ID [2107]
Internal file name [OUTPUT/2107_Sunday_February_25_2024_06_49_50_AM_3166317/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: 35.
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]]

\[ \boxed {12 y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }-3 y^{\prime \prime }+y^{\prime }=0} \] The characteristic equation is \[ 12 \lambda ^{4}-4 \lambda ^{3}-3 \lambda ^{2}+\lambda = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= {\frac {1}{3}}\\ \lambda _3 &= -{\frac {1}{2}}\\ \lambda _4 &= {\frac {1}{2}} \end {align*}

Therefore the homogeneous solution is \[ y_h(x)=c_{1} +{\mathrm e}^{\frac {x}{3}} c_{2} +{\mathrm e}^{-\frac {x}{2}} c_{3} +{\mathrm e}^{\frac {x}{2}} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= 1\\ y_2 &= {\mathrm e}^{\frac {x}{3}}\\ y_3 &= {\mathrm e}^{-\frac {x}{2}}\\ y_4 &= {\mathrm e}^{\frac {x}{2}} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} +{\mathrm e}^{\frac {x}{3}} c_{2} +{\mathrm e}^{-\frac {x}{2}} c_{3} +{\mathrm e}^{\frac {x}{2}} c_{4} \\ \end{align*}

Verification of solutions

\[ y = c_{1} +{\mathrm e}^{\frac {x}{3}} c_{2} +{\mathrm e}^{-\frac {x}{2}} c_{3} +{\mathrm e}^{\frac {x}{2}} c_{4} \] Verified OK.

9.21.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 12 y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }-3 y^{\prime \prime }+y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & y^{\prime \prime \prime \prime } \\ \bullet & {} & \textrm {Isolate 4th derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime \prime }=\frac {y^{\prime \prime \prime }}{3}+\frac {y^{\prime \prime }}{4}-\frac {y^{\prime }}{12} \\ \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 \prime }-\frac {y^{\prime \prime \prime }}{3}-\frac {y^{\prime \prime }}{4}+\frac {y^{\prime }}{12}=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 {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 )=\frac {y_{4}\left (x \right )}{3}+\frac {y_{3}\left (x \right )}{4}-\frac {y_{2}\left (x \right )}{12} \\ & {} & \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 )=\frac {y_{4}\left (x \right )}{3}+\frac {y_{3}\left (x \right )}{4}-\frac {y_{2}\left (x \right )}{12}\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 \\ 0 & -\frac {1}{12} & \frac {1}{4} & \frac {1}{3} \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 \\ 0 & -\frac {1}{12} & \frac {1}{4} & \frac {1}{3} \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} -8 \\ 4 \\ -2 \\ 1 \end {array}\right ]\right ], \left [0, \left [\begin {array}{c} 1 \\ 0 \\ 0 \\ 0 \end {array}\right ]\right ], \left [\frac {1}{3}, \left [\begin {array}{c} 27 \\ 9 \\ 3 \\ 1 \end {array}\right ]\right ], \left [\frac {1}{2}, \left [\begin {array}{c} 8 \\ 4 \\ 2 \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {1}{2}, \left [\begin {array}{c} -8 \\ 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} -8 \\ 4 \\ -2 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [0, \left [\begin {array}{c} 1 \\ 0 \\ 0 \\ 0 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{2}=\left [\begin {array}{c} 1 \\ 0 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [\frac {1}{3}, \left [\begin {array}{c} 27 \\ 9 \\ 3 \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{3}={\mathrm e}^{\frac {x}{3}}\cdot \left [\begin {array}{c} 27 \\ 9 \\ 3 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [\frac {1}{2}, \left [\begin {array}{c} 8 \\ 4 \\ 2 \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{4}={\mathrm e}^{\frac {x}{2}}\cdot \left [\begin {array}{c} 8 \\ 4 \\ 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}+c_{4} {\moverset {\rightarrow }{y}}_{4} \\ \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} -8 \\ 4 \\ -2 \\ 1 \end {array}\right ]+c_{3} {\mathrm e}^{\frac {x}{3}}\cdot \left [\begin {array}{c} 27 \\ 9 \\ 3 \\ 1 \end {array}\right ]+{\mathrm e}^{\frac {x}{2}} c_{4} \cdot \left [\begin {array}{c} 8 \\ 4 \\ 2 \\ 1 \end {array}\right ]+\left [\begin {array}{c} c_{2} \\ 0 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\left (8 \,{\mathrm e}^{x} c_{4} +27 c_{3} {\mathrm e}^{\frac {5 x}{6}}+c_{2} {\mathrm e}^{\frac {x}{2}}-8 c_{1} \right ) {\mathrm e}^{-\frac {x}{2}} \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: 0.0 (sec). Leaf size: 27

dsolve(12*diff(y(x),x$4)-4*diff(y(x),x$3)-3*diff(y(x),x$2)+diff(y(x),x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = \left (c_{2} {\mathrm e}^{x}+c_{3} {\mathrm e}^{\frac {5 x}{6}}+c_{1} {\mathrm e}^{\frac {x}{2}}+c_{4} \right ) {\mathrm e}^{-\frac {x}{2}} \]

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 39

DSolve[12*y''''[x]-4*y'''[x]-3*y''[x]+y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-x/2} \left (3 c_1 e^{5 x/6}+2 c_3 e^x-2 c_2\right )+c_4 \]