Internal problem ID [12347]
Internal file name [OUTPUT/11000_Monday_October_02_2023_02_47_47_AM_80377253/index.tex]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A.
Dobrushkin. CRC Press 2015
Section: Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page
368
Problem number: Problem 13(d).
ODE order: 3.
ODE degree: 1.
The type(s) of ODE detected by this program : "higher_order_laplace"
Maple gives the following as the ode type
[[_3rd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime \prime }-5 y^{\prime \prime }+y^{\prime }-y=-t^{2}+2 t -10} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 0] \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 ) \end {align*}
The given ode becomes an algebraic equation in the Laplace domain \[ s^{3} Y \left (s \right )-y^{\prime \prime }\left (0\right )-s y^{\prime }\left (0\right )-s^{2} y \left (0\right )-5 s^{2} Y \left (s \right )+5 y^{\prime }\left (0\right )+5 s y \left (0\right )+s Y \left (s \right )-y \left (0\right )-Y \left (s \right ) = -\frac {2}{s^{3}}+\frac {2}{s^{2}}-\frac {10}{s}\tag {1} \] But the initial conditions are \begin {align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=0 \end {align*}
Substituting these initial conditions in above in Eq (1) gives \[ s^{3} Y \left (s \right )-2-2 s^{2}-5 s^{2} Y \left (s \right )+10 s +s Y \left (s \right )-Y \left (s \right ) = -\frac {2}{s^{3}}+\frac {2}{s^{2}}-\frac {10}{s} \] Solving the above equation for \(Y(s)\) results in \[ Y(s) = \frac {2 s^{5}-10 s^{4}+2 s^{3}-10 s^{2}+2 s -2}{s^{3} \left (s^{3}-5 s^{2}+s -1\right )} \] Applying partial fractions decomposition results in \[ Y(s)= \frac {2}{s^{3}}+\frac {\frac {\left (\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{3}+\frac {22}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}+\frac {5}{3}\right )^{2}}{26}-\frac {11 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{78}-\frac {121}{39 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}+\frac {29}{78}}{s -\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{3}-\frac {22}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}-\frac {5}{3}}+\frac {\frac {\left (-\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{6}-\frac {11}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{3}-\frac {22}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}}{26}+\frac {11 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{156}+\frac {121}{78 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}+\frac {29}{78}-\frac {11 i \sqrt {3}\, \left (\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{3}-\frac {22}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}\right )}{52}}{s +\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{6}+\frac {11}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}-\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{3}-\frac {22}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}\right )}{2}}+\frac {\frac {\left (-\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{6}-\frac {11}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{3}-\frac {22}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}}{26}+\frac {11 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{156}+\frac {121}{78 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}+\frac {29}{78}+\frac {11 i \sqrt {3}\, \left (\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{3}-\frac {22}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}\right )}{52}}{s +\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{6}+\frac {11}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}-\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{3}-\frac {22}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}\right )}{2}} \] The inverse Laplace of each term above is now found, which gives \begin {align*} \mathcal {L}^{-1}\left (\frac {2}{s^{3}}\right ) &= t^{2}\\ \mathcal {L}^{-1}\left (\frac {\frac {\left (\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{3}+\frac {22}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}+\frac {5}{3}\right )^{2}}{26}-\frac {11 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{78}-\frac {121}{39 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}+\frac {29}{78}}{s -\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{3}-\frac {22}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}-\frac {5}{3}}\right ) &= \frac {{\mathrm e}^{\frac {\left (-3 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}} \sqrt {78}+58 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}+242 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}+1210\right ) t}{726}} \left (-2184+157 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}-138 \sqrt {78}-506 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}\right )}{234 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}}\\ \mathcal {L}^{-1}\left (\frac {\frac {\left (-\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{6}-\frac {11}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{3}-\frac {22}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}}{26}+\frac {11 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{156}+\frac {121}{78 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}+\frac {29}{78}-\frac {11 i \sqrt {3}\, \left (\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{3}-\frac {22}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}\right )}{52}}{s +\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{6}+\frac {11}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}-\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{3}-\frac {22}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}\right )}{2}}\right ) &= \frac {\left (-\left (116+6 \sqrt {78}\right )^{\frac {4}{3}}+312 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}-i \sqrt {3}\, \left (116+6 \sqrt {78}\right )^{\frac {4}{3}}-46 i \left (58+3 \sqrt {78}\right ) \sqrt {3}+2184+506 i \sqrt {3}\, \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}+138 \sqrt {78}+484 i \sqrt {3}+506 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}\right ) \left (\cos \left (\frac {\left (3 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}} \sqrt {78}\, \sqrt {3}-58 \sqrt {3}\, \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}+242 \sqrt {3}\, \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}\right ) t}{1452}\right )+i \sin \left (\frac {\left (3 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}} \sqrt {78}\, \sqrt {3}-58 \sqrt {3}\, \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}+242 \sqrt {3}\, \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}\right ) t}{1452}\right )\right ) {\mathrm e}^{\frac {\left (3 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}} \sqrt {78}-58 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}-242 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}+2420\right ) t}{1452}}}{468 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}}\\ \mathcal {L}^{-1}\left (\frac {\frac {\left (-\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{6}-\frac {11}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{3}-\frac {22}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}}{26}+\frac {11 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{156}+\frac {121}{78 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}+\frac {29}{78}+\frac {11 i \sqrt {3}\, \left (\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{3}-\frac {22}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}\right )}{52}}{s +\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{6}+\frac {11}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}-\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}{3}-\frac {22}{3 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}}\right )}{2}}\right ) &= \frac {\left (-\left (116+6 \sqrt {78}\right )^{\frac {4}{3}}+312 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}+i \sqrt {3}\, \left (116+6 \sqrt {78}\right )^{\frac {4}{3}}+46 i \left (58+3 \sqrt {78}\right ) \sqrt {3}+2184-506 i \sqrt {3}\, \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}+138 \sqrt {78}-484 i \sqrt {3}+506 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}\right ) \left (\cos \left (\frac {\left (-3 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}} \sqrt {78}\, \sqrt {3}+58 \sqrt {3}\, \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}-242 \sqrt {3}\, \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}\right ) t}{1452}\right )+i \sin \left (\frac {\left (-3 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}} \sqrt {78}\, \sqrt {3}+58 \sqrt {3}\, \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}-242 \sqrt {3}\, \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}\right ) t}{1452}\right )\right ) {\mathrm e}^{\frac {\left (3 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}} \sqrt {78}-58 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}-242 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}+2420\right ) t}{1452}}}{468 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}} \end {align*}
Adding the above results and simplifying gives \[ y=t^{2}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{3}-5 \textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-11 \underline {\hspace {1.25 ex}}\alpha +28\right ) {\mathrm e}^{\underline {\hspace {1.25 ex}}\alpha t}\right )}{26} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= t^{2}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha &=\operatorname {RootOf}\left (\textit {\_Z}^{3}-5 \textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-11 \underline {\hspace {1.25 ex}}\alpha +28\right ) {\mathrm e}^{\underline {\hspace {1.25 ex}}\alpha t}\right )}{26} \\ \end{align*}
Verification of solutions
\[ y = t^{2}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{3}-5 \textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-11 \underline {\hspace {1.25 ex}}\alpha +28\right ) {\mathrm e}^{\underline {\hspace {1.25 ex}}\alpha t}\right )}{26} \] Verified OK.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 3; linear nonhomogeneous with symmetry [0,1] trying high order linear exact nonhomogeneous trying differential order: 3; missing the dependent variable checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 4.391 (sec). Leaf size: 38
dsolve([diff(y(t),t$3)-5*diff(y(t),t$2)+diff(y(t),t)-y(t)=2*t-10-t^2,y(0) = 2, D(y)(0) = 0, (D@@2)(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = \frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{3}-5 \textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\left (\underline {\hspace {1.25 ex}}\alpha -4\right ) \left (\underline {\hspace {1.25 ex}}\alpha -7\right ) {\mathrm e}^{\underline {\hspace {1.25 ex}}\alpha t}\right )}{26}+t^{2} \]
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 1009
DSolve[{y'''[t]-5*y''[t]+y'[t]-y[t]==2*t-10-t^2,{y[0]==2,y'[0]==0,y''[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {-\text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ] \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ]^2 t^2+\text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ] \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ]^2 t^2+\text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ]^2 \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ] t^2-\text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ]^2 \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ] t^2-\text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ] \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ]^2 t^2+\text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ]^2 \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ] t^2-2 e^{t \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ]} \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ] \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ]^2+2 e^{t \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ]} \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ] \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ]^2+2 e^{t \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ]} \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ]-2 e^{t \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ]} \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ]+2 e^{t \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ]} \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ]^2 \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ]-2 e^{t \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ]} \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ]^2 \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ]-2 e^{t \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ]} \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ] \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ]^2-2 e^{t \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ]} \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ]+2 e^{t \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ]} \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ]+2 e^{t \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ]} \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ]^2 \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ]+2 e^{t \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ]} \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ]-2 e^{t \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ]} \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ]}{\left (\text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ]\right ) \left (-\text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ]+\text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ]\right ) \left (-\text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ]\right )} \]