Internal problem ID [5048]
Internal file name [OUTPUT/4541_Sunday_June_05_2022_03_00_34_PM_44292396/index.tex]
Book: Basic Training in Mathematics. By R. Shankar. Plenum Press. NY. 1995
Section: Chapter 10, Differential equations. Section 10.2, ODEs with constant Coefficients. page
307
Problem number: 10.2.8 part(2).
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]]
\[ \boxed {x^{\prime \prime \prime \prime }+x=0} \] The characteristic equation is \[ \lambda ^{4}+1 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\\ \lambda _2 &= -\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\\ \lambda _3 &= -\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\\ \lambda _4 &= \frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2} \end {align*}
Therefore the homogeneous solution is \[ x_h(t)={\mathrm e}^{\left (-\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) t} c_{1} +{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) t} c_{2} +{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) t} c_{3} +{\mathrm e}^{\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) t} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} x_1 &= {\mathrm e}^{\left (-\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) t}\\ x_2 &= {\mathrm e}^{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) t}\\ x_3 &= {\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) t}\\ x_4 &= {\mathrm e}^{\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) t} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} x &= {\mathrm e}^{\left (-\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) t} c_{1} +{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) t} c_{2} +{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) t} c_{3} +{\mathrm e}^{\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) t} c_{4} \\ \end{align*}
Verification of solutions
\[ x = {\mathrm e}^{\left (-\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) t} c_{1} +{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) t} c_{2} +{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) t} c_{3} +{\mathrm e}^{\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) t} c_{4} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime \prime \prime \prime }+x=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & x^{\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} x_{1}\left (t \right ) \\ {} & {} & x_{1}\left (t \right )=x \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} x_{2}\left (t \right ) \\ {} & {} & x_{2}\left (t \right )=x^{\prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} x_{3}\left (t \right ) \\ {} & {} & x_{3}\left (t \right )=x^{\prime \prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} x_{4}\left (t \right ) \\ {} & {} & x_{4}\left (t \right )=x^{\prime \prime \prime } \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} x_{4}^{\prime }\left (t \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & x_{4}^{\prime }\left (t \right )=-x_{1}\left (t \right ) \\ & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & {} & \left [x_{2}\left (t \right )=x_{1}^{\prime }\left (t \right ), x_{3}\left (t \right )=x_{2}^{\prime }\left (t \right ), x_{4}\left (t \right )=x_{3}^{\prime }\left (t \right ), x_{4}^{\prime }\left (t \right )=-x_{1}\left (t \right )\right ] \\ \bullet & {} & \textrm {Define vector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}\left (t \right )=\left [\begin {array}{c} x_{1}\left (t \right ) \\ x_{2}\left (t \right ) \\ x_{3}\left (t \right ) \\ x_{4}\left (t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}^{\prime }\left (t \right )=\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \end {array}\right ]\cdot {\moverset {\rightarrow }{x}}\left (t \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 \\ -1 & 0 & 0 & 0 \end {array}\right ] \\ \bullet & {} & \textrm {Rewrite the system as}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}^{\prime }\left (t \right )=A \cdot {\moverset {\rightarrow }{x}}\left (t \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 {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{3}} \\ \frac {1}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{2}} \\ \frac {1}{-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}} \\ 1 \end {array}\right ]\right ], \left [-\frac {\sqrt {2}}{2}+\frac {\mathrm {I} \sqrt {2}}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\sqrt {2}}{2}+\frac {\mathrm {I} \sqrt {2}}{2}\right )^{3}} \\ \frac {1}{\left (-\frac {\sqrt {2}}{2}+\frac {\mathrm {I} \sqrt {2}}{2}\right )^{2}} \\ \frac {1}{-\frac {\sqrt {2}}{2}+\frac {\mathrm {I} \sqrt {2}}{2}} \\ 1 \end {array}\right ]\right ], \left [\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{3}} \\ \frac {1}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{2}} \\ \frac {1}{\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}} \\ 1 \end {array}\right ]\right ], \left [\frac {\sqrt {2}}{2}+\frac {\mathrm {I} \sqrt {2}}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {2}}{2}+\frac {\mathrm {I} \sqrt {2}}{2}\right )^{3}} \\ \frac {1}{\left (\frac {\sqrt {2}}{2}+\frac {\mathrm {I} \sqrt {2}}{2}\right )^{2}} \\ \frac {1}{\frac {\sqrt {2}}{2}+\frac {\mathrm {I} \sqrt {2}}{2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{3}} \\ \frac {1}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{2}} \\ \frac {1}{-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right ) t}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{3}} \\ \frac {1}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{2}} \\ \frac {1}{-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}} \\ 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 {\sqrt {2}\, t}{2}}\cdot \left (\cos \left (\frac {\sqrt {2}\, t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, t}{2}\right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{3}} \\ \frac {1}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{2}} \\ \frac {1}{-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{3}} \\ \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{2}} \\ \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}} \\ \cos \left (\frac {\sqrt {2}\, t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{x}}_{1}\left (t \right )={\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ -\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \\ -\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ \cos \left (\frac {\sqrt {2}\, t}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{x}}_{2}\left (t \right )={\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}-\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ -\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \\ \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ -\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{3}} \\ \frac {1}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{2}} \\ \frac {1}{\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right ) t}\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{3}} \\ \frac {1}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{2}} \\ \frac {1}{\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}} \\ 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 {\sqrt {2}\, t}{2}}\cdot \left (\cos \left (\frac {\sqrt {2}\, t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, t}{2}\right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{3}} \\ \frac {1}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{2}} \\ \frac {1}{\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{3}} \\ \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}\right )^{2}} \\ \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}} \\ \cos \left (\frac {\sqrt {2}\, t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{x}}_{3}\left (t \right )={\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\cdot \left [\begin {array}{c} -\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \\ \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ \cos \left (\frac {\sqrt {2}\, t}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{x}}_{4}\left (t \right )={\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ \cos \left (\frac {\sqrt {2}\, t}{2}\right ) \\ \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}-\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ -\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {General solution to the system of ODEs}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}=c_{1} {\moverset {\rightarrow }{x}}_{1}\left (t \right )+c_{2} {\moverset {\rightarrow }{x}}_{2}\left (t \right )+c_{3} {\moverset {\rightarrow }{x}}_{3}\left (t \right )+c_{4} {\moverset {\rightarrow }{x}}_{4}\left (t \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}=c_{1} {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ -\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \\ -\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ \cos \left (\frac {\sqrt {2}\, t}{2}\right ) \end {array}\right ]+c_{2} {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}-\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ -\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \\ \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ -\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \end {array}\right ]+c_{3} {\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\cdot \left [\begin {array}{c} -\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \\ \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ \cos \left (\frac {\sqrt {2}\, t}{2}\right ) \end {array}\right ]+c_{4} {\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ \cos \left (\frac {\sqrt {2}\, t}{2}\right ) \\ \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}-\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ -\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & x=\frac {\left (\left (\left (c_{1} +c_{2} \right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (c_{3} -c_{4} \right )\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \left (\left (c_{1} -c_{2} \right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}+{\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (c_{3} +c_{4} \right )\right )\right ) \sqrt {2}}{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: 0.015 (sec). Leaf size: 61
dsolve(diff(x(t),t$4)+x(t)=0,x(t), singsol=all)
\[ x \left (t \right ) = \left (-c_{1} {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-c_{2} {\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \sin \left (\frac {\sqrt {2}\, t}{2}\right )+\left (c_{3} {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}+c_{4} {\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 65
DSolve[x''''[t]+x[t]==0,x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to e^{-\frac {t}{\sqrt {2}}} \left (\left (c_1 e^{\sqrt {2} t}+c_2\right ) \cos \left (\frac {t}{\sqrt {2}}\right )+\left (c_4 e^{\sqrt {2} t}+c_3\right ) \sin \left (\frac {t}{\sqrt {2}}\right )\right ) \]