Internal problem ID [12293]
Internal file name [OUTPUT/10946_Saturday_September_30_2023_08_26_34_PM_40367264/index.tex]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A.
Dobrushkin. CRC Press 2015
Section: Chapter 5.5 Laplace transform. Homogeneous equations. Problems page 357
Problem number: Problem 13.
ODE order: 4.
ODE degree: 1.
The type(s) of ODE detected by this program : "higher_order_laplace"
Maple gives the following as the ode type
[[_high_order, _missing_x]]
\[ \boxed {y^{\prime \prime \prime \prime }+y=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 0, y^{\prime \prime \prime }\left (0\right ) = \frac {\sqrt {2}}{2}\right ] \end {align*}
Solving using the Laplace transform method. Let \[ \mathcal {L}\left (y\right ) =Y(s) \] Taking the Laplace transform of the ode and using the relations that \begin {align*} \mathcal {L}\left (y^{\prime }\right )&= s Y(s) - y \left (0\right )\\ \mathcal {L}\left (y^{\prime \prime }\right ) &= s^2 Y(s) - y'(0) - s y \left (0\right )\\ \mathcal {L}\left (y^{\prime \prime \prime }\right ) &= s^3 Y(s) - y''(0) - s y'(0) - s^2 y \left (0\right )\\ \mathcal {L}\left (y^{\prime \prime \prime \prime }\right ) &= s^4 Y(s) - y'''(0) - s y''(0) - s^2 y'(0)- s^3 y \left (0\right ) \end {align*}
The given ode becomes an algebraic equation in the Laplace domain \[ s^{4} Y \left (s \right )-y^{\prime \prime \prime }\left (0\right )-s y^{\prime \prime }\left (0\right )-s^{2} y^{\prime }\left (0\right )-s^{3} y \left (0\right )+Y \left (s \right ) = 0\tag {1} \] But the initial conditions are \begin {align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=0\\ y^{\prime \prime \prime }\left (0\right )&=\frac {\sqrt {2}}{2} \end {align*}
Substituting these initial conditions in above in Eq (1) gives \[ s^{4} Y \left (s \right )-\frac {\sqrt {2}}{2}-s^{3}+Y \left (s \right ) = 0 \] Solving the above equation for \(Y(s)\) results in \[ Y(s) = \frac {2 s^{3}+\sqrt {2}}{2 s^{4}+2} \] Applying partial fractions decomposition results in \[ Y(s)= \frac {\frac {1}{4}-\frac {\sqrt {2}\, \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{8}}{s -\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}+\frac {\frac {1}{4}-\frac {\sqrt {2}\, \left (-\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{8}}{s +\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}+\frac {\frac {1}{4}-\frac {\sqrt {2}\, \left (-\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right )}{8}}{s +\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}+\frac {\frac {1}{4}-\frac {\sqrt {2}\, \left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right )}{8}}{s -\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}} \] The inverse Laplace of each term above is now found, which gives \begin {align*} \mathcal {L}^{-1}\left (\frac {\frac {1}{4}-\frac {\sqrt {2}\, \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{8}}{s -\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}\right ) &= \left (\frac {1}{8}-\frac {i}{8}\right ) {\mathrm e}^{\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, t}\\ \mathcal {L}^{-1}\left (\frac {\frac {1}{4}-\frac {\sqrt {2}\, \left (-\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{8}}{s +\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}\right ) &= \left (\frac {3}{8}-\frac {i}{8}\right ) {\mathrm e}^{\left (-\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, t}\\ \mathcal {L}^{-1}\left (\frac {\frac {1}{4}-\frac {\sqrt {2}\, \left (-\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right )}{8}}{s +\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}\right ) &= \left (\frac {3}{8}+\frac {i}{8}\right ) {\mathrm e}^{\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, t}\\ \mathcal {L}^{-1}\left (\frac {\frac {1}{4}-\frac {\sqrt {2}\, \left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right )}{8}}{s -\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}\right ) &= \left (\frac {1}{8}+\frac {i}{8}\right ) {\mathrm e}^{\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, t} \end {align*}
Adding the above results and simplifying gives \[ y=-\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sinh \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\cosh \left (\frac {\sqrt {2}\, t}{2}\right ) \left (2 \cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sin \left (\frac {\sqrt {2}\, t}{2}\right )\right )}{2} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sinh \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\cosh \left (\frac {\sqrt {2}\, t}{2}\right ) \left (2 \cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sin \left (\frac {\sqrt {2}\, t}{2}\right )\right )}{2} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \sinh \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\cosh \left (\frac {\sqrt {2}\, t}{2}\right ) \left (2 \cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sin \left (\frac {\sqrt {2}\, t}{2}\right )\right )}{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime \prime \prime \prime }+y=0, y \left (0\right )=1, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=0, y^{\prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=0, y^{\prime \prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=\frac {\sqrt {2}}{2}\right ] \\ \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 (t \right ) \\ {} & {} & y_{1}\left (t \right )=y \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{2}\left (t \right ) \\ {} & {} & y_{2}\left (t \right )=y^{\prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{3}\left (t \right ) \\ {} & {} & y_{3}\left (t \right )=y^{\prime \prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{4}\left (t \right ) \\ {} & {} & y_{4}\left (t \right )=y^{\prime \prime \prime } \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} y_{4}^{\prime }\left (t \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{4}^{\prime }\left (t \right )=-y_{1}\left (t \right ) \\ & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & {} & \left [y_{2}\left (t \right )=y_{1}^{\prime }\left (t \right ), y_{3}\left (t \right )=y_{2}^{\prime }\left (t \right ), y_{4}\left (t \right )=y_{3}^{\prime }\left (t \right ), y_{4}^{\prime }\left (t \right )=-y_{1}\left (t \right )\right ] \\ \bullet & {} & \textrm {Define vector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}\left (t \right )=\left [\begin {array}{c} y_{1}\left (t \right ) \\ y_{2}\left (t \right ) \\ y_{3}\left (t \right ) \\ y_{4}\left (t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\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 }{y}}\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 }{y}}^{\prime }\left (t \right )=A \cdot {\moverset {\rightarrow }{y}}\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 }{y}}_{1}\left (t \right )={\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}\cdot \left [\begin {array}{c} \frac {\sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{2} \\ -\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \\ -\frac {\sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{2} \\ \cos \left (\frac {\sqrt {2}\, t}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{2}\left (t \right )={\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}\cdot \left [\begin {array}{c} \frac {\sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}-\frac {\sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{2} \\ -\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \\ \frac {\sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{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 }{y}}_{3}\left (t \right )={\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\cdot \left [\begin {array}{c} -\frac {\sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{2} \\ \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \\ \frac {\sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{2} \\ \cos \left (\frac {\sqrt {2}\, t}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{4}\left (t \right )={\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\cdot \left [\begin {array}{c} \frac {\sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{2} \\ \cos \left (\frac {\sqrt {2}\, t}{2}\right ) \\ \frac {\sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}-\frac {\sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{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 }{y}}=c_{1} {\moverset {\rightarrow }{y}}_{1}\left (t \right )+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (t \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (t \right )+c_{4} {\moverset {\rightarrow }{y}}_{4}\left (t \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}=c_{1} {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}\cdot \left [\begin {array}{c} \frac {\sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{2} \\ -\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \\ -\frac {\sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{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 {\sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}-\frac {\sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{2} \\ -\cos \left (\frac {\sqrt {2}\, t}{2}\right ) \\ \frac {\sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{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 {\sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{2} \\ \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \\ \frac {\sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{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 {\sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{2} \\ \cos \left (\frac {\sqrt {2}\, t}{2}\right ) \\ \frac {\sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}-\frac {\sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{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} \\ {} & {} & y=\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 )+\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 ) \sin \left (\frac {\sqrt {2}\, t}{2}\right )\right ) \sqrt {2}}{2} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y \left (0\right )=1 \\ {} & {} & 1=\frac {\left (c_{1} +c_{2} -c_{3} +c_{4} \right ) \sqrt {2}}{2} \\ \bullet & {} & \textrm {Calculate the 1st derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (\left (-\frac {\left (c_{1} +c_{2} \right ) \sqrt {2}\, {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}}{2}-\frac {\sqrt {2}\, {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (c_{3} -c_{4} \right )}{2}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )-\frac {\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 ) \sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\left (-\frac {\left (c_{1} -c_{2} \right ) \sqrt {2}\, {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}}{2}+\frac {\sqrt {2}\, {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (c_{3} +c_{4} \right )}{2}\right ) \sin \left (\frac {\sqrt {2}\, t}{2}\right )+\frac {\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 ) \sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}\right ) \sqrt {2}}{2} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=0 \\ {} & {} & 0=\frac {\left (-\frac {\left (c_{1} +c_{2} \right ) \sqrt {2}}{2}-\frac {\sqrt {2}\, \left (c_{3} -c_{4} \right )}{2}+\frac {\left (c_{1} -c_{2} +c_{3} +c_{4} \right ) \sqrt {2}}{2}\right ) \sqrt {2}}{2} \\ \bullet & {} & \textrm {Calculate the 2nd derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime \prime }=\frac {\left (\left (\frac {\left (c_{1} +c_{2} \right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}}{2}-\frac {{\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (c_{3} -c_{4} \right )}{2}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )-\left (-\frac {\left (c_{1} +c_{2} \right ) \sqrt {2}\, {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}}{2}-\frac {\sqrt {2}\, {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (c_{3} -c_{4} \right )}{2}\right ) \sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )-\frac {\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 )}{2}+\left (\frac {\left (c_{1} -c_{2} \right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}}{2}+\frac {{\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (c_{3} +c_{4} \right )}{2}\right ) \sin \left (\frac {\sqrt {2}\, t}{2}\right )+\left (-\frac {\left (c_{1} -c_{2} \right ) \sqrt {2}\, {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}}{2}+\frac {\sqrt {2}\, {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (c_{3} +c_{4} \right )}{2}\right ) \sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )-\frac {\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 ) \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{2}\right ) \sqrt {2}}{2} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y^{\prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=0 \\ {} & {} & 0=-\frac {\left (c_{1} -c_{2} \right ) \sqrt {2}}{2}+\frac {\sqrt {2}\, \left (c_{3} +c_{4} \right )}{2} \\ \bullet & {} & \textrm {Calculate the 3rd derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }=\frac {\left (\left (-\frac {\left (c_{1} +c_{2} \right ) \sqrt {2}\, {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}}{4}-\frac {\sqrt {2}\, {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (c_{3} -c_{4} \right )}{4}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )-\frac {3 \left (\frac {\left (c_{1} +c_{2} \right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}}{2}-\frac {{\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (c_{3} -c_{4} \right )}{2}\right ) \sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{2}-\frac {3 \left (-\frac {\left (c_{1} +c_{2} \right ) \sqrt {2}\, {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}}{2}-\frac {\sqrt {2}\, {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (c_{3} -c_{4} \right )}{2}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\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 ) \sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{4}+\left (-\frac {\left (c_{1} -c_{2} \right ) \sqrt {2}\, {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}}{4}+\frac {\sqrt {2}\, {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (c_{3} +c_{4} \right )}{4}\right ) \sin \left (\frac {\sqrt {2}\, t}{2}\right )+\frac {3 \left (\frac {\left (c_{1} -c_{2} \right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}}{2}+\frac {{\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (c_{3} +c_{4} \right )}{2}\right ) \sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}-\frac {3 \left (-\frac {\left (c_{1} -c_{2} \right ) \sqrt {2}\, {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}}{2}+\frac {\sqrt {2}\, {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \left (c_{3} +c_{4} \right )}{2}\right ) \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{2}-\frac {\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 ) \sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{4}\right ) \sqrt {2}}{2} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y^{\prime \prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=\frac {\sqrt {2}}{2} \\ {} & {} & \frac {\sqrt {2}}{2}=\frac {\left (\frac {\left (c_{1} +c_{2} \right ) \sqrt {2}}{2}+\frac {\sqrt {2}\, \left (c_{3} -c_{4} \right )}{2}+\frac {3 \left (\frac {c_{1}}{2}-\frac {c_{2}}{2}+\frac {c_{3}}{2}+\frac {c_{4}}{2}\right ) \sqrt {2}}{2}-\frac {\left (c_{1} -c_{2} +c_{3} +c_{4} \right ) \sqrt {2}}{4}\right ) \sqrt {2}}{2} \\ \bullet & {} & \textrm {Solve for the unknown coefficients}\hspace {3pt} \\ {} & {} & \left \{c_{1} =\frac {\sqrt {2}}{2}, c_{2} =\frac {\sqrt {2}}{4}, c_{3} =0, c_{4} =\frac {\sqrt {2}}{4}\right \} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {\left (3 \,{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}+{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{4}+\frac {\left ({\mathrm e}^{-\frac {\sqrt {2}\, t}{2}}+{\mathrm e}^{\frac {\sqrt {2}\, t}{2}}\right ) \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{4} \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: 4.985 (sec). Leaf size: 47
dsolve([diff(y(t),t$4)+y(t)=0,y(0) = 1, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 1/sqrt(2)],y(t), singsol=all)
\[ y \left (t \right ) = -\frac {\sinh \left (\frac {\sqrt {2}\, t}{2}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\cosh \left (\frac {\sqrt {2}\, t}{2}\right ) \left (2 \cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sin \left (\frac {\sqrt {2}\, t}{2}\right )\right )}{2} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 61
DSolve[{y''''[t]+y[t]==0,{y[0]==0,y'[0]==0,y''[0]==0,y'''[0]==1/Sqrt[2]}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {1}{4} e^{-\frac {t}{\sqrt {2}}} \left (\left (e^{\sqrt {2} t}+1\right ) \sin \left (\frac {t}{\sqrt {2}}\right )-\left (e^{\sqrt {2} t}-1\right ) \cos \left (\frac {t}{\sqrt {2}}\right )\right ) \]