Internal problem ID [11517]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag,
NY. 2015.
Section: Chapter 2, Second order linear equations. Section 2.5 Higher order equations. Exercises
page 130
Problem number: 2.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _missing_x]]
\[ \boxed {x^{\prime \prime \prime }+x^{\prime \prime }-x^{\prime }-4 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 1, x^{\prime }\left (0\right ) = 0, x^{\prime \prime }\left (0\right ) = -1] \end {align*}
✓ Solution by Maple
Time used: 0.516 (sec). Leaf size: 296
dsolve([diff(x(t),t$3)+diff(x(t),t$2)-diff(x(t),t)-4*x(t)=0,x(0) = 1, D(x)(0) = 0, (D@@2)(x)(0) = -1],x(t), singsol=all)
\[ x \left (t \right ) = \frac {\left (\left (\left (32 \sqrt {113}+352\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\left (-\sqrt {113}-25\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}+776 \sqrt {113}+8136\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-16\right ) t}{12 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )+32 \left (\left (\sqrt {113}+11\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\frac {\left (388+36 \sqrt {113}\right )^{\frac {2}{3}} \left (\sqrt {113}+25\right )}{32}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-16\right ) t}{12 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\right ) {\mathrm e}^{-\frac {t \left (4+\frac {\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}}{4}+\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}\right )}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}}-32 \,{\mathrm e}^{-\frac {t \left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}}{2}+\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}-8\right )}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}} \left (\left (\sqrt {113}+11\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\left (-\frac {\sqrt {113}}{32}-\frac {25}{32}\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-\frac {97 \sqrt {113}}{8}-\frac {1017}{8}\right )}{1164 \sqrt {113}+12204} \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 748
DSolve[{x'''[t]+x''[t]-x'[t]-4*x[t]==0,{x[0]==1,x'[0]==0,x''[0]==-1}},x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \frac {\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,1\right ] \exp \left (t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,2\right ]\right )-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,1\right ] \exp \left (t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,3\right ]\right )-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,2\right ] \exp \left (t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,1\right ]\right )+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,1\right ]^2 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,2\right ] \exp \left (t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,3\right ]\right )-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,1\right ] \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,2\right ]^2 \exp \left (t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,3\right ]\right )+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,3\right ] \exp \left (t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,1\right ]\right )-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,1\right ]^2 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,3\right ] \exp \left (t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,2\right ]\right )+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,2\right ]^2 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,3\right ] \exp \left (t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,1\right ]\right )+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,1\right ] \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,3\right ]^2 \exp \left (t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,2\right ]\right )-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,2\right ] \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,3\right ]^2 \exp \left (t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,1\right ]\right )+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,2\right ] \exp \left (t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,3\right ]\right )-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,3\right ] \exp \left (t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,2\right ]\right )}{\left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,3\right ]-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,2\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,2\right ]\right ) \left (-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,1\right ]+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}-4\&,3\right ]\right )} \]