Internal problem ID [14644]
Internal file name [OUTPUT/14324_Wednesday_April_03_2024_02_17_19_PM_78487314/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 4. Higher Order Equations. Exercises 4.5, page 175
Problem number: 26.
ODE order: 3.
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
[[_3rd_order, _missing_x]]
\[ \boxed {5 y^{\prime \prime \prime }-15 y^{\prime }+11 y=0} \] The characteristic equation is \[ 5 \lambda ^{3}-15 \lambda +11 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\\ \lambda _2 &= \frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= \frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}
Therefore the homogeneous solution is \[ y_h(t)={\mathrm e}^{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{1} +{\mathrm e}^{\left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t} c_{2} +{\mathrm e}^{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\\ y_2 &= {\mathrm e}^{\left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}\\ y_3 &= {\mathrm e}^{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{1} +{\mathrm e}^{\left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t} c_{2} +{\mathrm e}^{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{3} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{1} +{\mathrm e}^{\left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t} c_{2} +{\mathrm e}^{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 5 y^{\prime \prime \prime }-15 y^{\prime }+11 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & y^{\prime \prime \prime } \\ \bullet & {} & \textrm {Isolate 3rd derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }=3 y^{\prime }-\frac {11 y}{5} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }-3 y^{\prime }+\frac {11 y}{5}=0 \\ \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 {Isolate for}\hspace {3pt} y_{3}^{\prime }\left (t \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{3}^{\prime }\left (t \right )=3 y_{2}\left (t \right )-\frac {11 y_{1}\left (t \right )}{5} \\ & {} & \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_{3}^{\prime }\left (t \right )=3 y_{2}\left (t \right )-\frac {11 y_{1}\left (t \right )}{5}\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 ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (t \right )=\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -\frac {11}{5} & 3 & 0 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (t \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -\frac {11}{5} & 3 & 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 {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )^{2}} \\ \frac {1}{-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}} \\ 1 \end {array}\right ]\right ], \left [\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ]\right ], \left [\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )^{2}} \\ \frac {1}{-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{\left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )^{2}} \\ \frac {1}{-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{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}^{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}\cdot \left (\cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}{2}\right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}{2}\right )}{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {\cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}{2}\right )}{\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) 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}}_{2}\left (t \right )={\mathrm e}^{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}\cdot \left [\begin {array}{c} \frac {50 \left (1100+100 \sqrt {21}\right )^{\frac {2}{3}} \left (\left (1100+100 \sqrt {21}\right )^{\frac {4}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )-\left (1100+100 \sqrt {21}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )+400 \left (1100+100 \sqrt {21}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )-10000 \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-10000 \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )\right )}{\left (\left (1100+100 \sqrt {21}\right )^{\frac {4}{3}}-100 \left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}+10000\right )^{2}} \\ \frac {5 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}} \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )+\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )-100 \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+100 \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )\right )}{\left (1100+100 \sqrt {21}\right )^{\frac {4}{3}}-100 \left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}+10000} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{3}\left (t \right )={\mathrm e}^{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}\cdot \left [\begin {array}{c} -\frac {50 \left (1100+100 \sqrt {21}\right )^{\frac {2}{3}} \left (\left (1100+100 \sqrt {21}\right )^{\frac {4}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )+\left (1100+100 \sqrt {21}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )-400 \left (1100+100 \sqrt {21}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )-10000 \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+10000 \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )\right )}{\left (\left (1100+100 \sqrt {21}\right )^{\frac {4}{3}}-100 \left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}+10000\right )^{2}} \\ -\frac {5 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}} \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )-\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )-100 \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-100 \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )\right )}{\left (1100+100 \sqrt {21}\right )^{\frac {4}{3}}-100 \left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}+10000} \\ -\sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) 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}+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (t \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (t \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}=c_{1} {\mathrm e}^{\left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )^{2}} \\ \frac {1}{-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}\cdot \left [\begin {array}{c} \frac {50 \left (1100+100 \sqrt {21}\right )^{\frac {2}{3}} \left (\left (1100+100 \sqrt {21}\right )^{\frac {4}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )-\left (1100+100 \sqrt {21}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )+400 \left (1100+100 \sqrt {21}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )-10000 \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-10000 \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )\right )}{\left (\left (1100+100 \sqrt {21}\right )^{\frac {4}{3}}-100 \left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}+10000\right )^{2}} \\ \frac {5 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}} \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )+\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )-100 \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+100 \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )\right )}{\left (1100+100 \sqrt {21}\right )^{\frac {4}{3}}-100 \left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}+10000} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ]+c_{3} {\mathrm e}^{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}\cdot \left [\begin {array}{c} -\frac {50 \left (1100+100 \sqrt {21}\right )^{\frac {2}{3}} \left (\left (1100+100 \sqrt {21}\right )^{\frac {4}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )+\left (1100+100 \sqrt {21}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )-400 \left (1100+100 \sqrt {21}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )-10000 \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+10000 \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )\right )}{\left (\left (1100+100 \sqrt {21}\right )^{\frac {4}{3}}-100 \left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}+10000\right )^{2}} \\ -\frac {5 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}} \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )-\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )-100 \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-100 \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )\right )}{\left (1100+100 \sqrt {21}\right )^{\frac {4}{3}}-100 \left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}+10000} \\ -\sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {3 \left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {2}{3}} \left ({\mathrm e}^{\frac {3 \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}+100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}} \left (\frac {\left (\frac {\left (-11 \sqrt {7}\, c_{2} +29 c_{3} \right ) \sqrt {3}}{3}-11 \sqrt {7}\, c_{3} +\frac {29 c_{2}}{3}\right ) \left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {2}{3}}}{100}+c_{2} \left (\sqrt {3}\, \sqrt {7}+11\right ) \left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {1}{3}}+\frac {2 \left (11 \sqrt {7}\, c_{2} -46 c_{3} \right ) \sqrt {3}}{3}+\frac {92 c_{2}}{3}-22 \sqrt {7}\, c_{3} \right ) \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {1}{3}}}\right )+\left (\frac {\left (\frac {\left (-11 \sqrt {7}\, c_{3} -29 c_{2} \right ) \sqrt {3}}{3}+11 \sqrt {7}\, c_{2} +\frac {29 c_{3}}{3}\right ) \left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {2}{3}}}{100}+c_{3} \left (\sqrt {3}\, \sqrt {7}+11\right ) \left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {1}{3}}+\frac {2 \left (11 \sqrt {7}\, c_{3} +46 c_{2} \right ) \sqrt {3}}{3}+\frac {92 c_{3}}{3}+22 \sqrt {7}\, c_{2} \right ) {\mathrm e}^{\frac {3 \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}+100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {1}{3}}}\right )+\left (-\frac {184}{3}+\frac {\left (11 \sqrt {3}\, \sqrt {7}-29\right ) \left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {2}{3}}}{150}+\left (\sqrt {3}\, \sqrt {7}+11\right ) \left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {1}{3}}-\frac {44 \sqrt {3}\, \sqrt {7}}{3}\right ) c_{1} \right ) {\mathrm e}^{-\frac {\left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}+100\right ) t}{10 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}}}{968 \left (11 \sqrt {3}\, \sqrt {7}+71\right )} \end {array} \]
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 149
dsolve(5*diff(y(t),t$3)-15*diff(y(t),t)+11*y(t)=0,y(t), singsol=all)
\[ y \left (t \right ) = \left (c_{2} {\mathrm e}^{\frac {3 \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}+100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {1}{3}}}\right )+c_{3} {\mathrm e}^{\frac {3 \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}+100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {1}{3}}}\right )+c_{1} \right ) {\mathrm e}^{-\frac {\left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}+100\right ) t}{10 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}} \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 75
DSolve[5*y'''[t]-15*y'[t]+11*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to c_2 \exp \left (t \text {Root}\left [5 \text {$\#$1}^3-15 \text {$\#$1}+11\&,2\right ]\right )+c_3 \exp \left (t \text {Root}\left [5 \text {$\#$1}^3-15 \text {$\#$1}+11\&,3\right ]\right )+c_1 \exp \left (t \text {Root}\left [5 \text {$\#$1}^3-15 \text {$\#$1}+11\&,1\right ]\right ) \]