Internal problem ID [14102]
Internal file name [OUTPUT/13783_Saturday_March_02_2024_02_50_31_PM_85608383/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number: 81.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }+\frac {25 y^{\prime \prime }}{2}-5 y^{\prime }+\frac {629 y}{16}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1, y^{\prime \prime }\left (0\right ) = -1, y^{\prime \prime \prime }\left (0\right ) = 1] \end {align*}
The characteristic equation is \[ -5 \lambda +\frac {25}{2} \lambda ^{2}+\lambda ^{4}+\frac {629}{16} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -\frac {1}{2}-3 i\\ \lambda _2 &= -\frac {1}{2}+3 i\\ \lambda _3 &= \frac {1}{2}-2 i\\ \lambda _4 &= \frac {1}{2}+2 i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (-\frac {1}{2}+3 i\right ) x} c_{1} +{\mathrm e}^{\left (\frac {1}{2}+2 i\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {1}{2}-3 i\right ) x} c_{3} +{\mathrm e}^{\left (\frac {1}{2}-2 i\right ) x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{\left (-\frac {1}{2}+3 i\right ) x}\\ y_2 &= {\mathrm e}^{\left (\frac {1}{2}+2 i\right ) x}\\ y_3 &= {\mathrm e}^{\left (-\frac {1}{2}-3 i\right ) x}\\ y_4 &= {\mathrm e}^{\left (\frac {1}{2}-2 i\right ) x} \end {align*}
Initial conditions are used to solve for the constants of integration.
Looking at the above solution \begin {align*} y = {\mathrm e}^{\left (-\frac {1}{2}+3 i\right ) x} c_{1} +{\mathrm e}^{\left (\frac {1}{2}+2 i\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {1}{2}-3 i\right ) x} c_{3} +{\mathrm e}^{\left (\frac {1}{2}-2 i\right ) x} c_{4} \tag {1} \end {align*}
Initial conditions are now substituted in the above solution. This will generate the required equations to solve for the integration constants. substituting \(y = 0\) and \(x = 0\) in the above gives \begin {align*} 0 = c_{1} +c_{2} +c_{3} +c_{4}\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} y^{\prime } = \left (-\frac {1}{2}+3 i\right ) {\mathrm e}^{\left (-\frac {1}{2}+3 i\right ) x} c_{1} +\left (\frac {1}{2}+2 i\right ) {\mathrm e}^{\left (\frac {1}{2}+2 i\right ) x} c_{2} +\left (-\frac {1}{2}-3 i\right ) {\mathrm e}^{\left (-\frac {1}{2}-3 i\right ) x} c_{3} +\left (\frac {1}{2}-2 i\right ) {\mathrm e}^{\left (\frac {1}{2}-2 i\right ) x} c_{4} \end {align*}
substituting \(y^{\prime } = 1\) and \(x = 0\) in the above gives \begin {align*} 1 = \left (-\frac {1}{2}+3 i\right ) c_{1} +\left (\frac {1}{2}+2 i\right ) c_{2} +\left (-\frac {1}{2}-3 i\right ) c_{3} +\left (\frac {1}{2}-2 i\right ) c_{4}\tag {2A} \end {align*}
Taking two derivatives of the solution gives \begin {align*} y^{\prime \prime } = \left (-\frac {35}{4}-3 i\right ) {\mathrm e}^{\left (-\frac {1}{2}+3 i\right ) x} c_{1} +\left (-\frac {15}{4}+2 i\right ) {\mathrm e}^{\left (\frac {1}{2}+2 i\right ) x} c_{2} +\left (-\frac {35}{4}+3 i\right ) {\mathrm e}^{\left (-\frac {1}{2}-3 i\right ) x} c_{3} +\left (-\frac {15}{4}-2 i\right ) {\mathrm e}^{\left (\frac {1}{2}-2 i\right ) x} c_{4} \end {align*}
substituting \(y^{\prime \prime } = -1\) and \(x = 0\) in the above gives \begin {align*} -1 = \left (-\frac {35}{4}-3 i\right ) c_{1} +\left (-\frac {15}{4}+2 i\right ) c_{2} +\left (-\frac {35}{4}+3 i\right ) c_{3} +\left (-\frac {15}{4}-2 i\right ) c_{4}\tag {3A} \end {align*}
Taking three derivatives of the solution gives \begin {align*} y^{\prime \prime \prime } = \left (\frac {107}{8}-\frac {99 i}{4}\right ) {\mathrm e}^{\left (-\frac {1}{2}+3 i\right ) x} c_{1} +\left (-\frac {47}{8}-\frac {13 i}{2}\right ) {\mathrm e}^{\left (\frac {1}{2}+2 i\right ) x} c_{2} +\left (\frac {107}{8}+\frac {99 i}{4}\right ) {\mathrm e}^{\left (-\frac {1}{2}-3 i\right ) x} c_{3} +\left (-\frac {47}{8}+\frac {13 i}{2}\right ) {\mathrm e}^{\left (\frac {1}{2}-2 i\right ) x} c_{4} \end {align*}
substituting \(y^{\prime \prime \prime } = 1\) and \(x = 0\) in the above gives \begin {align*} 1 = \left (\frac {107}{8}-\frac {99 i}{4}\right ) c_{1} +\left (-\frac {47}{8}-\frac {13 i}{2}\right ) c_{2} +\left (\frac {107}{8}+\frac {99 i}{4}\right ) c_{3} +\left (-\frac {47}{8}+\frac {13 i}{2}\right ) c_{4}\tag {4A} \end {align*}
Equations {1A,2A,3A,4A} are now solved for \(\{c_{1}, c_{2}, c_{3}, c_{4}\}\). Solving for the constants gives \begin {align*} c_{1}&=\frac {37}{208}-\frac {5 i}{104}\\ c_{2}&=-\frac {37}{208}-\frac {111 i}{416}\\ c_{3}&=\frac {37}{208}+\frac {5 i}{104}\\ c_{4}&=-\frac {37}{208}+\frac {111 i}{416} \end {align*}
Substituting these values back in above solution results in \begin {align*} y = \frac {37 \,{\mathrm e}^{\left (-\frac {1}{2}+3 i\right ) x}}{208}-\frac {5 i {\mathrm e}^{\left (-\frac {1}{2}+3 i\right ) x}}{104}-\frac {37 \,{\mathrm e}^{\left (\frac {1}{2}+2 i\right ) x}}{208}-\frac {111 i {\mathrm e}^{\left (\frac {1}{2}+2 i\right ) x}}{416}+\frac {37 \,{\mathrm e}^{\left (-\frac {1}{2}-3 i\right ) x}}{208}+\frac {5 i {\mathrm e}^{\left (-\frac {1}{2}-3 i\right ) x}}{104}-\frac {37 \,{\mathrm e}^{\left (\frac {1}{2}-2 i\right ) x}}{208}+\frac {111 i {\mathrm e}^{\left (\frac {1}{2}-2 i\right ) x}}{416} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \left (\frac {37}{208}+\frac {5 i}{104}\right ) {\mathrm e}^{\left (-\frac {1}{2}-3 i\right ) x}+\left (\frac {37}{208}-\frac {5 i}{104}\right ) {\mathrm e}^{\left (-\frac {1}{2}+3 i\right ) x}+\left (-\frac {37}{208}+\frac {111 i}{416}\right ) {\mathrm e}^{\left (\frac {1}{2}-2 i\right ) x}+\left (-\frac {37}{208}-\frac {111 i}{416}\right ) {\mathrm e}^{\left (\frac {1}{2}+2 i\right ) x} \\ \end{align*}
Verification of solutions
\[ y = \left (\frac {37}{208}+\frac {5 i}{104}\right ) {\mathrm e}^{\left (-\frac {1}{2}-3 i\right ) x}+\left (\frac {37}{208}-\frac {5 i}{104}\right ) {\mathrm e}^{\left (-\frac {1}{2}+3 i\right ) x}+\left (-\frac {37}{208}+\frac {111 i}{416}\right ) {\mathrm e}^{\left (\frac {1}{2}-2 i\right ) x}+\left (-\frac {37}{208}-\frac {111 i}{416}\right ) {\mathrm e}^{\left (\frac {1}{2}+2 i\right ) x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d x}\frac {d}{d x}y^{\prime \prime }+\frac {25 \frac {d}{d x}y^{\prime }}{2}-5 y^{\prime }+\frac {629 y}{16}=0, y \left (0\right )=0, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=1, \left (\frac {d}{d x}y^{\prime }\right )\bigg | {\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=-1, \left (\frac {d}{d x}y^{\prime \prime }\right )\bigg | {\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=1\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\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 )=\frac {d}{d x}y^{\prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{4}\left (x \right ) \\ {} & {} & y_{4}\left (x \right )=\frac {d}{d x}y^{\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 {25 y_{3}\left (x \right )}{2}+5 y_{2}\left (x \right )-\frac {629 y_{1}\left (x \right )}{16} \\ & {} & \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 {25 y_{3}\left (x \right )}{2}+5 y_{2}\left (x \right )-\frac {629 y_{1}\left (x \right )}{16}\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 \\ -\frac {629}{16} & 5 & -\frac {25}{2} & 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 \\ -\frac {629}{16} & 5 & -\frac {25}{2} & 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 [-\frac {1}{2}-3 \,\mathrm {I}, \left [\begin {array}{c} \frac {856}{50653}-\frac {1584 \,\mathrm {I}}{50653} \\ -\frac {140}{1369}-\frac {48 \,\mathrm {I}}{1369} \\ -\frac {2}{37}+\frac {12 \,\mathrm {I}}{37} \\ 1 \end {array}\right ]\right ], \left [-\frac {1}{2}+3 \,\mathrm {I}, \left [\begin {array}{c} \frac {856}{50653}+\frac {1584 \,\mathrm {I}}{50653} \\ -\frac {140}{1369}+\frac {48 \,\mathrm {I}}{1369} \\ -\frac {2}{37}-\frac {12 \,\mathrm {I}}{37} \\ 1 \end {array}\right ]\right ], \left [\frac {1}{2}-2 \,\mathrm {I}, \left [\begin {array}{c} -\frac {376}{4913}-\frac {416 \,\mathrm {I}}{4913} \\ -\frac {60}{289}+\frac {32 \,\mathrm {I}}{289} \\ \frac {2}{17}+\frac {8 \,\mathrm {I}}{17} \\ 1 \end {array}\right ]\right ], \left [\frac {1}{2}+2 \,\mathrm {I}, \left [\begin {array}{c} -\frac {376}{4913}+\frac {416 \,\mathrm {I}}{4913} \\ -\frac {60}{289}-\frac {32 \,\mathrm {I}}{289} \\ \frac {2}{17}-\frac {8 \,\mathrm {I}}{17} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-\frac {1}{2}-3 \,\mathrm {I}, \left [\begin {array}{c} \frac {856}{50653}-\frac {1584 \,\mathrm {I}}{50653} \\ -\frac {140}{1369}-\frac {48 \,\mathrm {I}}{1369} \\ -\frac {2}{37}+\frac {12 \,\mathrm {I}}{37} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-\frac {1}{2}-3 \,\mathrm {I}\right ) x}\cdot \left [\begin {array}{c} \frac {856}{50653}-\frac {1584 \,\mathrm {I}}{50653} \\ -\frac {140}{1369}-\frac {48 \,\mathrm {I}}{1369} \\ -\frac {2}{37}+\frac {12 \,\mathrm {I}}{37} \\ 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 (3 x \right )-\mathrm {I} \sin \left (3 x \right )\right )\cdot \left [\begin {array}{c} \frac {856}{50653}-\frac {1584 \,\mathrm {I}}{50653} \\ -\frac {140}{1369}-\frac {48 \,\mathrm {I}}{1369} \\ -\frac {2}{37}+\frac {12 \,\mathrm {I}}{37} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-\frac {x}{2}}\cdot \left [\begin {array}{c} \left (\frac {856}{50653}-\frac {1584 \,\mathrm {I}}{50653}\right ) \left (\cos \left (3 x \right )-\mathrm {I} \sin \left (3 x \right )\right ) \\ \left (-\frac {140}{1369}-\frac {48 \,\mathrm {I}}{1369}\right ) \left (\cos \left (3 x \right )-\mathrm {I} \sin \left (3 x \right )\right ) \\ \left (-\frac {2}{37}+\frac {12 \,\mathrm {I}}{37}\right ) \left (\cos \left (3 x \right )-\mathrm {I} \sin \left (3 x \right )\right ) \\ \cos \left (3 x \right )-\mathrm {I} \sin \left (3 x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{1}\left (x \right )={\mathrm e}^{-\frac {x}{2}}\cdot \left [\begin {array}{c} \frac {856 \cos \left (3 x \right )}{50653}-\frac {1584 \sin \left (3 x \right )}{50653} \\ -\frac {140 \cos \left (3 x \right )}{1369}-\frac {48 \sin \left (3 x \right )}{1369} \\ -\frac {2 \cos \left (3 x \right )}{37}+\frac {12 \sin \left (3 x \right )}{37} \\ \cos \left (3 x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{2}\left (x \right )={\mathrm e}^{-\frac {x}{2}}\cdot \left [\begin {array}{c} -\frac {856 \sin \left (3 x \right )}{50653}-\frac {1584 \cos \left (3 x \right )}{50653} \\ \frac {140 \sin \left (3 x \right )}{1369}-\frac {48 \cos \left (3 x \right )}{1369} \\ \frac {2 \sin \left (3 x \right )}{37}+\frac {12 \cos \left (3 x \right )}{37} \\ -\sin \left (3 x \right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [\frac {1}{2}-2 \,\mathrm {I}, \left [\begin {array}{c} -\frac {376}{4913}-\frac {416 \,\mathrm {I}}{4913} \\ -\frac {60}{289}+\frac {32 \,\mathrm {I}}{289} \\ \frac {2}{17}+\frac {8 \,\mathrm {I}}{17} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {1}{2}-2 \,\mathrm {I}\right ) x}\cdot \left [\begin {array}{c} -\frac {376}{4913}-\frac {416 \,\mathrm {I}}{4913} \\ -\frac {60}{289}+\frac {32 \,\mathrm {I}}{289} \\ \frac {2}{17}+\frac {8 \,\mathrm {I}}{17} \\ 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 (2 x \right )-\mathrm {I} \sin \left (2 x \right )\right )\cdot \left [\begin {array}{c} -\frac {376}{4913}-\frac {416 \,\mathrm {I}}{4913} \\ -\frac {60}{289}+\frac {32 \,\mathrm {I}}{289} \\ \frac {2}{17}+\frac {8 \,\mathrm {I}}{17} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\frac {x}{2}}\cdot \left [\begin {array}{c} \left (-\frac {376}{4913}-\frac {416 \,\mathrm {I}}{4913}\right ) \left (\cos \left (2 x \right )-\mathrm {I} \sin \left (2 x \right )\right ) \\ \left (-\frac {60}{289}+\frac {32 \,\mathrm {I}}{289}\right ) \left (\cos \left (2 x \right )-\mathrm {I} \sin \left (2 x \right )\right ) \\ \left (\frac {2}{17}+\frac {8 \,\mathrm {I}}{17}\right ) \left (\cos \left (2 x \right )-\mathrm {I} \sin \left (2 x \right )\right ) \\ \cos \left (2 x \right )-\mathrm {I} \sin \left (2 x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{3}\left (x \right )={\mathrm e}^{\frac {x}{2}}\cdot \left [\begin {array}{c} -\frac {376 \cos \left (2 x \right )}{4913}-\frac {416 \sin \left (2 x \right )}{4913} \\ -\frac {60 \cos \left (2 x \right )}{289}+\frac {32 \sin \left (2 x \right )}{289} \\ \frac {2 \cos \left (2 x \right )}{17}+\frac {8 \sin \left (2 x \right )}{17} \\ \cos \left (2 x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{4}\left (x \right )={\mathrm e}^{\frac {x}{2}}\cdot \left [\begin {array}{c} \frac {376 \sin \left (2 x \right )}{4913}-\frac {416 \cos \left (2 x \right )}{4913} \\ \frac {60 \sin \left (2 x \right )}{289}+\frac {32 \cos \left (2 x \right )}{289} \\ -\frac {2 \sin \left (2 x \right )}{17}+\frac {8 \cos \left (2 x \right )}{17} \\ -\sin \left (2 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}\left (x \right )+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (x \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (x \right )+c_{4} {\moverset {\rightarrow }{y}}_{4}\left (x \right ) \\ \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} \frac {856 \cos \left (3 x \right )}{50653}-\frac {1584 \sin \left (3 x \right )}{50653} \\ -\frac {140 \cos \left (3 x \right )}{1369}-\frac {48 \sin \left (3 x \right )}{1369} \\ -\frac {2 \cos \left (3 x \right )}{37}+\frac {12 \sin \left (3 x \right )}{37} \\ \cos \left (3 x \right ) \end {array}\right ]+c_{2} {\mathrm e}^{-\frac {x}{2}}\cdot \left [\begin {array}{c} -\frac {856 \sin \left (3 x \right )}{50653}-\frac {1584 \cos \left (3 x \right )}{50653} \\ \frac {140 \sin \left (3 x \right )}{1369}-\frac {48 \cos \left (3 x \right )}{1369} \\ \frac {2 \sin \left (3 x \right )}{37}+\frac {12 \cos \left (3 x \right )}{37} \\ -\sin \left (3 x \right ) \end {array}\right ]+c_{3} {\mathrm e}^{\frac {x}{2}}\cdot \left [\begin {array}{c} -\frac {376 \cos \left (2 x \right )}{4913}-\frac {416 \sin \left (2 x \right )}{4913} \\ -\frac {60 \cos \left (2 x \right )}{289}+\frac {32 \sin \left (2 x \right )}{289} \\ \frac {2 \cos \left (2 x \right )}{17}+\frac {8 \sin \left (2 x \right )}{17} \\ \cos \left (2 x \right ) \end {array}\right ]+c_{4} {\mathrm e}^{\frac {x}{2}}\cdot \left [\begin {array}{c} \frac {376 \sin \left (2 x \right )}{4913}-\frac {416 \cos \left (2 x \right )}{4913} \\ \frac {60 \sin \left (2 x \right )}{289}+\frac {32 \cos \left (2 x \right )}{289} \\ -\frac {2 \sin \left (2 x \right )}{17}+\frac {8 \cos \left (2 x \right )}{17} \\ -\sin \left (2 x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {8 \left (\left (107 c_{1} -198 c_{2} \right ) \cos \left (3 x \right )-198 \sin \left (3 x \right ) \left (c_{1} +\frac {107 c_{2}}{198}\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{50653}-\frac {376 \,{\mathrm e}^{\frac {x}{2}} \left (\left (c_{3} +\frac {52 c_{4}}{47}\right ) \cos \left (2 x \right )+\frac {52 \sin \left (2 x \right ) \left (c_{3} -\frac {47 c_{4}}{52}\right )}{47}\right )}{4913} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y \left (0\right )=0 \\ {} & {} & 0=\frac {856 c_{1}}{50653}-\frac {1584 c_{2}}{50653}-\frac {376 c_{3}}{4913}-\frac {416 c_{4}}{4913} \\ \bullet & {} & \textrm {Calculate the 1st derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {8 \left (-3 \left (107 c_{1} -198 c_{2} \right ) \sin \left (3 x \right )-594 \cos \left (3 x \right ) \left (c_{1} +\frac {107 c_{2}}{198}\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{50653}-\frac {4 \left (\left (107 c_{1} -198 c_{2} \right ) \cos \left (3 x \right )-198 \sin \left (3 x \right ) \left (c_{1} +\frac {107 c_{2}}{198}\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{50653}-\frac {188 \,{\mathrm e}^{\frac {x}{2}} \left (\left (c_{3} +\frac {52 c_{4}}{47}\right ) \cos \left (2 x \right )+\frac {52 \sin \left (2 x \right ) \left (c_{3} -\frac {47 c_{4}}{52}\right )}{47}\right )}{4913}-\frac {376 \,{\mathrm e}^{\frac {x}{2}} \left (-2 \left (c_{3} +\frac {52 c_{4}}{47}\right ) \sin \left (2 x \right )+\frac {104 \cos \left (2 x \right ) \left (c_{3} -\frac {47 c_{4}}{52}\right )}{47}\right )}{4913} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=1 \\ {} & {} & 1=-\frac {140 c_{1}}{1369}-\frac {48 c_{2}}{1369}-\frac {60 c_{3}}{289}+\frac {32 c_{4}}{289} \\ \bullet & {} & \textrm {Calculate the 2nd derivative of the solution}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=\frac {8 \left (-9 \left (107 c_{1} -198 c_{2} \right ) \cos \left (3 x \right )+1782 \sin \left (3 x \right ) \left (c_{1} +\frac {107 c_{2}}{198}\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{50653}-\frac {8 \left (-3 \left (107 c_{1} -198 c_{2} \right ) \sin \left (3 x \right )-594 \cos \left (3 x \right ) \left (c_{1} +\frac {107 c_{2}}{198}\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{50653}+\frac {2 \left (\left (107 c_{1} -198 c_{2} \right ) \cos \left (3 x \right )-198 \sin \left (3 x \right ) \left (c_{1} +\frac {107 c_{2}}{198}\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{50653}-\frac {94 \,{\mathrm e}^{\frac {x}{2}} \left (\left (c_{3} +\frac {52 c_{4}}{47}\right ) \cos \left (2 x \right )+\frac {52 \sin \left (2 x \right ) \left (c_{3} -\frac {47 c_{4}}{52}\right )}{47}\right )}{4913}-\frac {376 \,{\mathrm e}^{\frac {x}{2}} \left (-2 \left (c_{3} +\frac {52 c_{4}}{47}\right ) \sin \left (2 x \right )+\frac {104 \cos \left (2 x \right ) \left (c_{3} -\frac {47 c_{4}}{52}\right )}{47}\right )}{4913}-\frac {376 \,{\mathrm e}^{\frac {x}{2}} \left (-4 \left (c_{3} +\frac {52 c_{4}}{47}\right ) \cos \left (2 x \right )-\frac {208 \sin \left (2 x \right ) \left (c_{3} -\frac {47 c_{4}}{52}\right )}{47}\right )}{4913} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} \left (\frac {d}{d x}y^{\prime }\right )\bigg | {\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=-1 \\ {} & {} & -1=-\frac {2 c_{1}}{37}+\frac {12 c_{2}}{37}+\frac {2 c_{3}}{17}+\frac {8 c_{4}}{17} \\ \bullet & {} & \textrm {Calculate the 3rd derivative of the solution}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }=\frac {8 \left (27 \left (107 c_{1} -198 c_{2} \right ) \sin \left (3 x \right )+5346 \cos \left (3 x \right ) \left (c_{1} +\frac {107 c_{2}}{198}\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{50653}-\frac {12 \left (-9 \left (107 c_{1} -198 c_{2} \right ) \cos \left (3 x \right )+1782 \sin \left (3 x \right ) \left (c_{1} +\frac {107 c_{2}}{198}\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{50653}+\frac {6 \left (-3 \left (107 c_{1} -198 c_{2} \right ) \sin \left (3 x \right )-594 \cos \left (3 x \right ) \left (c_{1} +\frac {107 c_{2}}{198}\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{50653}-\frac {\left (\left (107 c_{1} -198 c_{2} \right ) \cos \left (3 x \right )-198 \sin \left (3 x \right ) \left (c_{1} +\frac {107 c_{2}}{198}\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{50653}-\frac {47 \,{\mathrm e}^{\frac {x}{2}} \left (\left (c_{3} +\frac {52 c_{4}}{47}\right ) \cos \left (2 x \right )+\frac {52 \sin \left (2 x \right ) \left (c_{3} -\frac {47 c_{4}}{52}\right )}{47}\right )}{4913}-\frac {282 \,{\mathrm e}^{\frac {x}{2}} \left (-2 \left (c_{3} +\frac {52 c_{4}}{47}\right ) \sin \left (2 x \right )+\frac {104 \cos \left (2 x \right ) \left (c_{3} -\frac {47 c_{4}}{52}\right )}{47}\right )}{4913}-\frac {564 \,{\mathrm e}^{\frac {x}{2}} \left (-4 \left (c_{3} +\frac {52 c_{4}}{47}\right ) \cos \left (2 x \right )-\frac {208 \sin \left (2 x \right ) \left (c_{3} -\frac {47 c_{4}}{52}\right )}{47}\right )}{4913}-\frac {376 \,{\mathrm e}^{\frac {x}{2}} \left (8 \left (c_{3} +\frac {52 c_{4}}{47}\right ) \sin \left (2 x \right )-\frac {416 \cos \left (2 x \right ) \left (c_{3} -\frac {47 c_{4}}{52}\right )}{47}\right )}{4913} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} \left (\frac {d}{d x}y^{\prime \prime }\right )\bigg | {\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=1 \\ {} & {} & 1=c_{1} +c_{3} \\ \bullet & {} & \textrm {Solve for the unknown coefficients}\hspace {3pt} \\ {} & {} & \left \{c_{1} =\frac {1979}{832}, c_{2} =-\frac {2099}{208}, c_{3} =-\frac {1147}{832}, c_{4} =\frac {9065}{1664}\right \} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {\left (74 \cos \left (3 x \right )+20 \sin \left (3 x \right )\right ) {\mathrm e}^{-\frac {x}{2}}}{208}-\frac {37 \,{\mathrm e}^{\frac {x}{2}} \left (\cos \left (2 x \right )-\frac {3 \sin \left (2 x \right )}{2}\right )}{104} \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: 0.062 (sec). Leaf size: 41
dsolve([diff(y(x),x$4)+25/2*diff(y(x),x$2)-5*diff(y(x),x)+629/16*y(x)=0,y(0) = 0, D(y)(0) = 1, (D@@2)(y)(0) = -1, (D@@3)(y)(0) = 1],y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (74 \cos \left (3 x \right )+20 \sin \left (3 x \right )\right ) {\mathrm e}^{-\frac {x}{2}}}{208}-\frac {37 \left (\cos \left (2 x \right )-\frac {3 \sin \left (2 x \right )}{2}\right ) {\mathrm e}^{\frac {x}{2}}}{104} \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 47
DSolve[{y''''[x]+25/2*y''[x]-5*y'[x]+629/16*y[x]==0,{y[0]==0,y'[0]==1,y''[0]==-1,y'''[0]==1}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{208} e^{-x/2} \left (111 e^x \sin (2 x)+20 \sin (3 x)-74 e^x \cos (2 x)+74 \cos (3 x)\right ) \]