10.19 problem 19

Internal problem ID [2135]
Internal file name [OUTPUT/2135_Monday_February_26_2024_09_17_47_AM_75656588/index.tex]

Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section: Exercise 18, page 82
Problem number: 19.
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]]

\[ \boxed {2 y^{\prime \prime \prime }-3 y^{\prime \prime }+10 y^{\prime }-15 y=0} \] The characteristic equation is \[ 2 \lambda ^{3}-3 \lambda ^{2}+10 \lambda -15 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= {\frac {3}{2}}\\ \lambda _2 &= i \sqrt {5}\\ \lambda _3 &= -i \sqrt {5} \end {align*}

Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{i \sqrt {5}\, x} c_{1} +{\mathrm e}^{-i \sqrt {5}\, x} c_{2} +{\mathrm e}^{\frac {3 x}{2}} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{i \sqrt {5}\, x}\\ y_2 &= {\mathrm e}^{-i \sqrt {5}\, x}\\ y_3 &= {\mathrm e}^{\frac {3 x}{2}} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{i \sqrt {5}\, x} c_{1} +{\mathrm e}^{-i \sqrt {5}\, x} c_{2} +{\mathrm e}^{\frac {3 x}{2}} c_{3} \\ \end{align*}

Verification of solutions

\[ y = {\mathrm e}^{i \sqrt {5}\, x} c_{1} +{\mathrm e}^{-i \sqrt {5}\, x} c_{2} +{\mathrm e}^{\frac {3 x}{2}} c_{3} \] Verified OK.

10.19.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 \frac {d}{d x}y^{\prime \prime }-3 \frac {d}{d x}y^{\prime }+10 y^{\prime }-15 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \\ \bullet & {} & \textrm {Isolate 3rd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }=\frac {3 \frac {d}{d x}y^{\prime }}{2}-5 y^{\prime }+\frac {15 y}{2} \\ \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} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }-\frac {3 \frac {d}{d x}y^{\prime }}{2}+5 y^{\prime }-\frac {15 y}{2}=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 )=\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 )=\frac {3 y_{3}\left (x \right )}{2}-5 y_{2}\left (x \right )+\frac {15 y_{1}\left (x \right )}{2} \\ & {} & \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 {3 y_{3}\left (x \right )}{2}-5 y_{2}\left (x \right )+\frac {15 y_{1}\left (x \right )}{2}\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 {15}{2} & -5 & \frac {3}{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 {15}{2} & -5 & \frac {3}{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 {3}{2}, \left [\begin {array}{c} \frac {4}{9} \\ \frac {2}{3} \\ 1 \end {array}\right ]\right ], \left [\mathrm {-I} \sqrt {5}, \left [\begin {array}{c} -\frac {1}{5} \\ \frac {\mathrm {I}}{5} \sqrt {5} \\ 1 \end {array}\right ]\right ], \left [\mathrm {I} \sqrt {5}, \left [\begin {array}{c} -\frac {1}{5} \\ -\frac {\mathrm {I}}{5} \sqrt {5} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [\frac {3}{2}, \left [\begin {array}{c} \frac {4}{9} \\ \frac {2}{3} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{\frac {3 x}{2}}\cdot \left [\begin {array}{c} \frac {4}{9} \\ \frac {2}{3} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [\mathrm {-I} \sqrt {5}, \left [\begin {array}{c} -\frac {1}{5} \\ \frac {\mathrm {I}}{5} \sqrt {5} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\mathrm {-I} \sqrt {5}\, x}\cdot \left [\begin {array}{c} -\frac {1}{5} \\ \frac {\mathrm {I}}{5} \sqrt {5} \\ 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 \\ {} & {} & \left (\cos \left (\sqrt {5}\, x \right )-\mathrm {I} \sin \left (\sqrt {5}\, x \right )\right )\cdot \left [\begin {array}{c} -\frac {1}{5} \\ \frac {\mathrm {I}}{5} \sqrt {5} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & \left [\begin {array}{c} -\frac {\cos \left (\sqrt {5}\, x \right )}{5}+\frac {\mathrm {I} \sin \left (\sqrt {5}\, x \right )}{5} \\ \frac {\mathrm {I}}{5} \left (\cos \left (\sqrt {5}\, x \right )-\mathrm {I} \sin \left (\sqrt {5}\, x \right )\right ) \sqrt {5} \\ \cos \left (\sqrt {5}\, x \right )-\mathrm {I} \sin \left (\sqrt {5}\, x \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 )=\left [\begin {array}{c} -\frac {\cos \left (\sqrt {5}\, x \right )}{5} \\ \frac {\sqrt {5}\, \sin \left (\sqrt {5}\, x \right )}{5} \\ \cos \left (\sqrt {5}\, x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{3}\left (x \right )=\left [\begin {array}{c} \frac {\sin \left (\sqrt {5}\, x \right )}{5} \\ \frac {\sqrt {5}\, \cos \left (\sqrt {5}\, x \right )}{5} \\ -\sin \left (\sqrt {5}\, x \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}}=c_{1} {\mathrm e}^{\frac {3 x}{2}}\cdot \left [\begin {array}{c} \frac {4}{9} \\ \frac {2}{3} \\ 1 \end {array}\right ]+\left [\begin {array}{c} \frac {c_{3} \sin \left (\sqrt {5}\, x \right )}{5}-\frac {c_{2} \cos \left (\sqrt {5}\, x \right )}{5} \\ \frac {c_{3} \sqrt {5}\, \cos \left (\sqrt {5}\, x \right )}{5}+\frac {c_{2} \sqrt {5}\, \sin \left (\sqrt {5}\, x \right )}{5} \\ -c_{3} \sin \left (\sqrt {5}\, x \right )+c_{2} \cos \left (\sqrt {5}\, x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {4 c_{1} {\mathrm e}^{\frac {3 x}{2}}}{9}+\frac {c_{3} \sin \left (\sqrt {5}\, x \right )}{5}-\frac {c_{2} \cos \left (\sqrt {5}\, x \right )}{5} \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: 27

dsolve(2*diff(y(x),x$3)-3*diff(y(x),x$2)+10*diff(y(x),x)-15*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {3 x}{2}}+c_{2} \sin \left (\sqrt {5}\, x \right )+c_{3} \cos \left (\sqrt {5}\, x \right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 38

DSolve[2*y'''[x]-3*y''[x]+10*y'[x]-15*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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