Internal problem ID [13618]
Internal file name [OUTPUT/12790_Saturday_February_17_2024_08_44_51_AM_64959173/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 19. Arbitrary Homogeneous linear equations with constant coefficients.
Additional exercises page 369
Problem number: 19.1 (c).
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]]
\[ \boxed {y^{\prime \prime \prime \prime }-34 y^{\prime \prime }+225 y=0} \] The characteristic equation is \[ \lambda ^{4}-34 \lambda ^{2}+225 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 5\\ \lambda _2 &= -5\\ \lambda _3 &= -3\\ \lambda _4 &= 3 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{-3 x} c_{1} +c_{2} {\mathrm e}^{3 x}+{\mathrm e}^{-5 x} c_{3} +{\mathrm e}^{5 x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-3 x}\\ y_2 &= {\mathrm e}^{3 x}\\ y_3 &= {\mathrm e}^{-5 x}\\ y_4 &= {\mathrm e}^{5 x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-3 x} c_{1} +c_{2} {\mathrm e}^{3 x}+{\mathrm e}^{-5 x} c_{3} +{\mathrm e}^{5 x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{-3 x} c_{1} +c_{2} {\mathrm e}^{3 x}+{\mathrm e}^{-5 x} c_{3} +{\mathrm e}^{5 x} c_{4} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime \prime }-34 y^{\prime \prime }+225 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & y^{\prime \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 {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 )=34 y_{3}\left (x \right )-225 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_{4}\left (x \right )=y_{3}^{\prime }\left (x \right ), y_{4}^{\prime }\left (x \right )=34 y_{3}\left (x \right )-225 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 ) \\ 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 \\ -225 & 0 & 34 & 0 \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 \\ -225 & 0 & 34 & 0 \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 [-5, \left [\begin {array}{c} -\frac {1}{125} \\ \frac {1}{25} \\ -\frac {1}{5} \\ 1 \end {array}\right ]\right ], \left [-3, \left [\begin {array}{c} -\frac {1}{27} \\ \frac {1}{9} \\ -\frac {1}{3} \\ 1 \end {array}\right ]\right ], \left [3, \left [\begin {array}{c} \frac {1}{27} \\ \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]\right ], \left [5, \left [\begin {array}{c} \frac {1}{125} \\ \frac {1}{25} \\ \frac {1}{5} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-5, \left [\begin {array}{c} -\frac {1}{125} \\ \frac {1}{25} \\ -\frac {1}{5} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{-5 x}\cdot \left [\begin {array}{c} -\frac {1}{125} \\ \frac {1}{25} \\ -\frac {1}{5} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-3, \left [\begin {array}{c} -\frac {1}{27} \\ \frac {1}{9} \\ -\frac {1}{3} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{2}={\mathrm e}^{-3 x}\cdot \left [\begin {array}{c} -\frac {1}{27} \\ \frac {1}{9} \\ -\frac {1}{3} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [3, \left [\begin {array}{c} \frac {1}{27} \\ \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{3}={\mathrm e}^{3 x}\cdot \left [\begin {array}{c} \frac {1}{27} \\ \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [5, \left [\begin {array}{c} \frac {1}{125} \\ \frac {1}{25} \\ \frac {1}{5} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{4}={\mathrm e}^{5 x}\cdot \left [\begin {array}{c} \frac {1}{125} \\ \frac {1}{25} \\ \frac {1}{5} \\ 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}^{-5 x}\cdot \left [\begin {array}{c} -\frac {1}{125} \\ \frac {1}{25} \\ -\frac {1}{5} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{-3 x}\cdot \left [\begin {array}{c} -\frac {1}{27} \\ \frac {1}{9} \\ -\frac {1}{3} \\ 1 \end {array}\right ]+c_{3} {\mathrm e}^{3 x}\cdot \left [\begin {array}{c} \frac {1}{27} \\ \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]+{\mathrm e}^{5 x} c_{4} \cdot \left [\begin {array}{c} \frac {1}{125} \\ \frac {1}{25} \\ \frac {1}{5} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {{\mathrm e}^{-5 x} \left ({\mathrm e}^{10 x} c_{4} -\frac {125 c_{2} {\mathrm e}^{2 x}}{27}+\frac {125 c_{3} {\mathrm e}^{8 x}}{27}-c_{1} \right )}{125} \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: 29
dsolve(diff(y(x),x$4)-34*diff(y(x),x$2)+225*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left ({\mathrm e}^{10 x} c_{1} +c_{3} {\mathrm e}^{8 x}+c_{2} {\mathrm e}^{2 x}+c_{4} \right ) {\mathrm e}^{-5 x} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 39
DSolve[y''''[x]-34*y''[x]+225*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-5 x} \left (e^{2 x} \left (c_3 e^{6 x}+c_4 e^{8 x}+c_2\right )+c_1\right ) \]