problem 1

Internal problem ID [17800]
Book : Diﬀerential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.2 (Basic Theory of First Order Linear Systems). Problems at page 398
Problem number : 1
Date solved : Tuesday, January 28, 2025 at 11:03:13 AM
CAS classiﬁcation : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }+5 y^{\prime \prime \prime }+4 y&=0 \end{align*}

\[ y = {\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{-\frac {4 t \left (\frac {\left (190+6 \sqrt {393}\right )^{{2}/{3}}}{4}+\left (190+6 \sqrt {393}\right )^{{1}/{3}}+7\right )}{3 \left (190+6 \sqrt {393}\right )^{{1}/{3}}}}+c_{3} {\mathrm e}^{\frac {\left (28+\left (190+6 \sqrt {393}\right )^{{2}/{3}}-8 \left (190+6 \sqrt {393}\right )^{{1}/{3}}\right ) t}{6 \left (190+6 \sqrt {393}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (190+6 \sqrt {3}\, \sqrt {131}\right )^{{2}/{3}}-28\right ) t}{6 \left (190+6 \sqrt {3}\, \sqrt {131}\right )^{{1}/{3}}}\right )+c_4 \,{\mathrm e}^{\frac {\left (28+\left (190+6 \sqrt {393}\right )^{{2}/{3}}-8 \left (190+6 \sqrt {393}\right )^{{1}/{3}}\right ) t}{6 \left (190+6 \sqrt {393}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (190+6 \sqrt {3}\, \sqrt {131}\right )^{{2}/{3}}-28\right ) t}{6 \left (190+6 \sqrt {3}\, \sqrt {131}\right )^{{1}/{3}}}\right ) \]

\[ y(t)\to c_2 \exp \left (t \text {Root}\left [\text {$\#$1}^3+4 \text {$\#$1}^2-4 \text {$\#$1}+4\&,2\right ]\right )+c_3 \exp \left (t \text {Root}\left [\text {$\#$1}^3+4 \text {$\#$1}^2-4 \text {$\#$1}+4\&,3\right ]\right )+c_1 \exp \left (t \text {Root}\left [\text {$\#$1}^3+4 \text {$\#$1}^2-4 \text {$\#$1}+4\&,1\right ]\right )+c_4 e^{-t} \]

76.24.1 problem 1