Internal problem ID [12196]
Internal file name [OUTPUT/10849_Thursday_September_21_2023_05_48_02_AM_3993535/index.tex]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER.
Problems page 172
Problem number: Problem 49.
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, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime \prime \prime }+x=t^{3}} \] This is higher order nonhomogeneous ODE. Let the solution be \[ x = x_h + x_p \] Where \(x_h\) is the solution to the homogeneous ODE And \(x_p\) is a particular solution to the nonhomogeneous ODE. \(x_h\) is the solution to \[ x^{\prime \prime \prime \prime }+x = 0 \] 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*} Now the particular solution to the given ODE is found \[ x^{\prime \prime \prime \prime }+x = t^{3} \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ t^{3} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{1, t, t^{2}, t^{3}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \left \{{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) t}, {\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) t}, {\mathrm e}^{\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) t}, {\mathrm e}^{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) t}\right \} \] Since there is no duplication between the basis function in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis in the UC_set. \[ x_p = A_{4} t^{3}+A_{3} t^{2}+A_{2} t +A_{1} \] The unknowns \(\{A_{1}, A_{2}, A_{3}, A_{4}\}\) are found by substituting the above trial solution \(x_p\) into the ODE and comparing coefficients. Substituting the trial solution into the ODE and simplifying gives \[ A_{4} t^{3}+A_{3} t^{2}+A_{2} t +A_{1} = t^{3} \] Solving for the unknowns by comparing coefficients results in \[ [A_{1} = 0, A_{2} = 0, A_{3} = 0, A_{4} = 1] \] Substituting the above back in the above trial solution \(x_p\), gives the particular solution \[ x_p = t^{3} \] Therefore the general solution is \begin{align*} x &= x_h + x_p \\ &= \left ({\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}\right ) + \left (t^{3}\right ) \\ \end{align*} Which simplifies to \[ x = {\mathrm e}^{\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, t} c_{1} +{\mathrm e}^{\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, t} c_{2} +{\mathrm e}^{\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, t} c_{3} +{\mathrm e}^{\left (-\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, t} c_{4} +t^{3} \]
The solution(s) found are the following \begin{align*} \tag{1} x &= {\mathrm e}^{\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, t} c_{1} +{\mathrm e}^{\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, t} c_{2} +{\mathrm e}^{\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, t} c_{3} +{\mathrm e}^{\left (-\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, t} c_{4} +t^{3} \\ \end{align*}
Verification of solutions
\[ x = {\mathrm e}^{\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, t} c_{1} +{\mathrm e}^{\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, t} c_{2} +{\mathrm e}^{\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, t} c_{3} +{\mathrm e}^{\left (-\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, t} c_{4} +t^{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime \prime \prime \prime }+x=t^{3} \\ \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 )=t^{3}-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 )=t^{3}-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 )+\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ t^{3} \end {array}\right ] \\ \bullet & {} & \textrm {Define the forcing function}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{f}}\left (t \right )=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ t^{3} \end {array}\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 )+{\moverset {\rightarrow }{f}} \\ \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 of the system of ODEs can be written in terms of the particular solution}\hspace {3pt} {\moverset {\rightarrow }{x}}_{p}\left (t \right ) \\ {} & {} & {\moverset {\rightarrow }{x}}\left (t \right )=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 )+{\moverset {\rightarrow }{x}}_{p}\left (t \right ) \\ \square & {} & \textrm {Fundamental matrix}\hspace {3pt} \\ {} & \circ & \textrm {Let}\hspace {3pt} \phi \left (t \right )\hspace {3pt}\textrm {be the matrix whose columns are the independent solutions of the homogeneous system.}\hspace {3pt} \\ {} & {} & \phi \left (t \right )=\left [\begin {array}{cccc} {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \left (\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}\right ) & {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \left (\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}-\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}\right ) & {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (-\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}\right ) & {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}\right ) \\ -{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) & -{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right ) & {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) & {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right ) \\ {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \left (-\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}\right ) & {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \left (\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}\right ) & {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}\right ) & {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}-\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}\right ) \\ {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right ) & -{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) & {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right ) & -{\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \end {array}\right ] \\ {} & \circ & \textrm {The fundamental matrix,}\hspace {3pt} \Phi \left (t \right )\hspace {3pt}\textrm {is a normalized version of}\hspace {3pt} \phi \left (t \right )\hspace {3pt}\textrm {satisfying}\hspace {3pt} \Phi \left (0\right )=I \hspace {3pt}\textrm {where}\hspace {3pt} I \hspace {3pt}\textrm {is the identity matrix}\hspace {3pt} \\ {} & {} & \Phi \left (t \right )=\phi \left (t \right )\cdot \frac {1}{\phi \left (0\right )} \\ {} & \circ & \textrm {Substitute the value of}\hspace {3pt} \phi \left (t \right )\hspace {3pt}\textrm {and}\hspace {3pt} \phi \left (0\right ) \\ {} & {} & \Phi \left (t \right )=\left [\begin {array}{cccc} {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \left (\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}\right ) & {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \left (\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}-\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}\right ) & {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (-\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}\right ) & {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}\right ) \\ -{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) & -{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right ) & {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) & {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right ) \\ {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \left (-\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}\right ) & {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \left (\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}\right ) & {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}\right ) & {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}-\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}\right ) \\ {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right ) & -{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) & {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right ) & -{\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \end {array}\right ]\cdot \frac {1}{\left [\begin {array}{cccc} \frac {\sqrt {2}}{2} & \frac {\sqrt {2}}{2} & -\frac {\sqrt {2}}{2} & \frac {\sqrt {2}}{2} \\ 0 & -1 & 0 & 1 \\ -\frac {\sqrt {2}}{2} & \frac {\sqrt {2}}{2} & \frac {\sqrt {2}}{2} & \frac {\sqrt {2}}{2} \\ 1 & 0 & 1 & 0 \end {array}\right ]} \\ {} & \circ & \textrm {Evaluate and simplify to get the fundamental matrix}\hspace {3pt} \\ {} & {} & \Phi \left (t \right )=\left [\begin {array}{cccc} \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}+{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right )}{2} & -\frac {\left (\left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )-\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}+{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right )\right ) \sqrt {2}}{4} & -\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right )}{2} & \frac {\left (\left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}+{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right )\right ) \sqrt {2}}{4} \\ -\frac {\left (\left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}+{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right )\right ) \sqrt {2}}{4} & \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}+{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right )}{2} & -\frac {\left (\left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )-\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}+{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right )\right ) \sqrt {2}}{4} & -\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right )}{2} \\ \frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right )}{2} & -\frac {\left (\left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}+{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right )\right ) \sqrt {2}}{4} & \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}+{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right )}{2} & -\frac {\left (\left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )-\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}+{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right )\right ) \sqrt {2}}{4} \\ \frac {\left (\left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )-\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}+{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right )\right ) \sqrt {2}}{4} & \frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right )}{2} & -\frac {\left (\left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}+{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right )\right ) \sqrt {2}}{4} & \frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}+{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right )}{2} \end {array}\right ] \\ \square & {} & \textrm {Find a particular solution of the system of ODEs using variation of parameters}\hspace {3pt} \\ {} & \circ & \textrm {Let the particular solution be the fundamental matrix multiplied by}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (t \right )\hspace {3pt}\textrm {and solve for}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (t \right ) \\ {} & {} & {\moverset {\rightarrow }{x}}_{p}\left (t \right )=\Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right ) \\ {} & \circ & \textrm {Take the derivative of the particular solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}_{p}^{\prime }\left (t \right )=\Phi ^{\prime }\left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right )+\Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (t \right ) \\ {} & \circ & \textrm {Substitute particular solution and its derivative into the system of ODEs}\hspace {3pt} \\ {} & {} & \Phi ^{\prime }\left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right )+\Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (t \right )=A \cdot \Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right )+{\moverset {\rightarrow }{f}}\left (t \right ) \\ {} & \circ & \textrm {The fundamental matrix has columns that are solutions to the homogeneous system so its derivative follows that of the homogeneous system}\hspace {3pt} \\ {} & {} & A \cdot \Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right )+\Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (t \right )=A \cdot \Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right )+{\moverset {\rightarrow }{f}}\left (t \right ) \\ {} & \circ & \textrm {Cancel like terms}\hspace {3pt} \\ {} & {} & \Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (t \right )={\moverset {\rightarrow }{f}}\left (t \right ) \\ {} & \circ & \textrm {Multiply by the inverse of the fundamental matrix}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{v}}^{\prime }\left (t \right )=\frac {1}{\Phi \left (t \right )}\cdot {\moverset {\rightarrow }{f}}\left (t \right ) \\ {} & \circ & \textrm {Integrate to solve for}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (t \right ) \\ {} & {} & {\moverset {\rightarrow }{v}}\left (t \right )=\int _{0}^{t}\frac {1}{\Phi \left (s \right )}\cdot {\moverset {\rightarrow }{f}}\left (s \right )d s \\ {} & \circ & \textrm {Plug}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (t \right )\hspace {3pt}\textrm {into the equation for the particular solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}_{p}\left (t \right )=\Phi \left (t \right )\cdot \left (\int _{0}^{t}\frac {1}{\Phi \left (s \right )}\cdot {\moverset {\rightarrow }{f}}\left (s \right )d s \right ) \\ {} & \circ & \textrm {Plug in the fundamental matrix and the forcing function and compute}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}_{p}\left (t \right )=\left [\begin {array}{c} -\frac {3 \sqrt {2}\, \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+t^{3}-\frac {3 \,{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}-\frac {3 \,{\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ \left (3 \,{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-3 \,{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \sin \left (\frac {\sqrt {2}\, t}{2}\right )+3 t^{2} \\ \frac {3 \sqrt {2}\, \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}-\frac {3 \,{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}-\frac {3 \,{\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+6 t \\ 6+\left (-3 \,{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-3 \,{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {Plug particular solution back into general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}\left (t \right )=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 )+\left [\begin {array}{c} -\frac {3 \sqrt {2}\, \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+t^{3}-\frac {3 \,{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}-\frac {3 \,{\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2} \\ \left (3 \,{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-3 \,{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \sin \left (\frac {\sqrt {2}\, t}{2}\right )+3 t^{2} \\ \frac {3 \sqrt {2}\, \left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}-\frac {3 \,{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}-\frac {3 \,{\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}}{2}+6 t \\ 6+\left (-3 \,{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-3 \,{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \cos \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 {\sqrt {2}\, \left (\left (c_{1} +c_{2} -3\right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}-{\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (c_{3} -c_{4} -3\right )\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}\, \left (c_{1} -c_{2} -3\right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}}{2}+\frac {\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}\, \left (c_{3} +c_{4} -3\right ) {\mathrm e}^{\frac {\sqrt {2}\, t}{2}}}{2}+t^{3} \end {array} \]
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 4; linear nonhomogeneous with symmetry [0,1] trying high order linear exact nonhomogeneous trying differential order: 4; missing the dependent variable checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 67
dsolve(diff(x(t),t$4)+x(t)=t^3,x(t), singsol=all)
\[ x \left (t \right ) = \left (c_{2} {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}+c_{4} {\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \sin \left (\frac {\sqrt {2}\, t}{2}\right )+t^{3}+c_{1} {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right )+c_{3} {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 78
DSolve[x''''[t]+x[t]==t^3,x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to e^{-\frac {t}{\sqrt {2}}} \left (e^{\frac {t}{\sqrt {2}}} t^3+\left (c_1 e^{\sqrt {2} t}+c_2\right ) \cos \left (\frac {t}{\sqrt {2}}\right )+\left (c_4 e^{\sqrt {2} t}+c_3\right ) \sin \left (\frac {t}{\sqrt {2}}\right )\right ) \]