Internal problem ID [11506]
Internal file name [OUTPUT/10489_Thursday_May_18_2023_04_20_55_AM_52111856/index.tex]
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.
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]]
\[ \boxed {x^{\prime \prime \prime }+x^{\prime \prime }-x^{\prime }-4 x=0} \] 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*}
The characteristic equation is \[ \lambda ^{3}+\lambda ^{2}-\lambda -4 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= \frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}+\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}\\ \lambda _2 &= -\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{12}-\frac {4}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}-\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= -\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{12}-\frac {4}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}-\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}
Therefore the homogeneous solution is \[ x_h(t)={\mathrm e}^{\left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{12}-\frac {4}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}-\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{1} +{\mathrm e}^{\left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}+\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t} c_{2} +{\mathrm e}^{\left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{12}-\frac {4}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}-\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} x_1 &= {\mathrm e}^{\left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{12}-\frac {4}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}-\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\\ x_2 &= {\mathrm e}^{\left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}+\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\\ x_3 &= {\mathrm e}^{\left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{12}-\frac {4}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}-\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} \end {align*}
Initial conditions are used to solve for the constants of integration.
Looking at the above solution \begin {align*} x = {\mathrm e}^{\left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{12}-\frac {4}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}-\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{1} +{\mathrm e}^{\left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}+\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t} c_{2} +{\mathrm e}^{\left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{12}-\frac {4}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}-\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{3} \tag {1} \end {align*}
Initial conditions are now substituted in the above solution. This will generate the required equations to solve for the integration constants. substituting \(x = 1\) and \(t = 0\) in the above gives \begin {align*} 1 = c_{1} +c_{2} +c_{3}\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} x^{\prime } = \left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{12}-\frac {4}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}-\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )}{2}\right ) {\mathrm e}^{\left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{12}-\frac {4}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}-\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{1} +\left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}+\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) {\mathrm e}^{\left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}+\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t} c_{2} +\left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{12}-\frac {4}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}-\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )}{2}\right ) {\mathrm e}^{\left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{12}-\frac {4}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}-\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{3} \end {align*}
substituting \(x^{\prime } = 0\) and \(t = 0\) in the above gives \begin {align*} 0 = \frac {\left (-i \left (c_{1} -c_{3} \right ) \sqrt {3}-c_{1} +2 c_{2} -c_{3} \right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-4 \left (c_{1} +c_{2} +c_{3} \right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}-16 i \left (-c_{1} +c_{3} \right ) \sqrt {3}-16 c_{1} +32 c_{2} -16 c_{3}}{12 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\tag {2A} \end {align*}
Taking two derivatives of the solution gives \begin {align*} x^{\prime \prime } = \left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{12}-\frac {4}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}-\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2} {\mathrm e}^{\left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{12}-\frac {4}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}-\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{1} +\left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}+\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2} {\mathrm e}^{\left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}+\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t} c_{2} +\left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{12}-\frac {4}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}-\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2} {\mathrm e}^{\left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{12}-\frac {4}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{6}-\frac {8}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{3} \end {align*}
substituting \(x^{\prime \prime } = -1\) and \(t = 0\) in the above gives \begin {align*} -1 = \frac {\left (i \left (c_{1} -c_{3} \right ) \sqrt {3}-c_{1} +2 c_{2} -c_{3} \right ) \left (\sqrt {113}+9\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+2 \left (c_{1} +c_{2} +c_{3} \right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}+4 \left (\sqrt {113}+9\right ) \left (i \left (c_{1} -c_{3} \right ) \sqrt {3}+c_{1} -2 c_{2} +c_{3} \right )}{2 \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}}\tag {3A} \end {align*}
Equations {1A,2A,3A} are now solved for \(\{c_{1}, c_{2}, c_{3}\}\). Solving for the constants gives \begin {align*} c_{1}&=\frac {4 \left (1682 \sqrt {3}\, \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+9 \sqrt {3}\, \left (388+36 \sqrt {113}\right )^{\frac {2}{3}} \sqrt {113}+21 i \left (388+36 \sqrt {113}\right )^{\frac {2}{3}} \sqrt {113}+162 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {113}+113 \sqrt {3}\, \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-96 i \left (388+36 \sqrt {113}\right )^{\frac {1}{3}} \sqrt {113}+189 i \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-1098 \sqrt {3}\, \sqrt {113}+966 i \sqrt {113}-1056 i \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}-11578 \sqrt {3}+10326 i\right ) \sqrt {3}}{9 \left (\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-16\right ) \left (\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-4 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+16\right ) \left (\sqrt {113}+9\right )}\\ c_{2}&=\frac {11 \left (388+36 \sqrt {113}\right )^{\frac {2}{3}} \sqrt {113}}{2712}-\frac {25 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}} \sqrt {113}}{5424}-\frac {\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}}{24}+\frac {\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}{48}+\frac {1}{3}\\ c_{3}&=\frac {4 \left (1682 \sqrt {3}\, \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+9 \sqrt {3}\, \left (388+36 \sqrt {113}\right )^{\frac {2}{3}} \sqrt {113}+162 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {113}+113 \sqrt {3}\, \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-1098 \sqrt {3}\, \sqrt {113}-21 i \left (388+36 \sqrt {113}\right )^{\frac {2}{3}} \sqrt {113}-11578 \sqrt {3}+96 i \left (388+36 \sqrt {113}\right )^{\frac {1}{3}} \sqrt {113}-189 i \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-966 i \sqrt {113}+1056 i \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}-10326 i\right ) \sqrt {3}}{9 \left (\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-16\right ) \left (\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-4 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+16\right ) \left (\sqrt {113}+9\right )} \end {align*}
Substituting these values back in above solution results in \begin {align*} x = \text {Expression too large to display} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} x &= \frac {\left (\left (2466225 i \sqrt {3}+232025 i \sqrt {339}-9438777-887841 \sqrt {113}\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\left (-317982 i \sqrt {3}-29902 i \sqrt {339}+61863 \sqrt {113}+657999\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}+20082360 i \sqrt {3}+1889144 i \sqrt {339}+6112920 \sqrt {113}+64982232\right ) {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-16 i \sqrt {3}+\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}+4 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+16\right ) t}{12 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}}+\left (\left (-2466225 i \sqrt {3}-232025 i \sqrt {339}-887841 \sqrt {113}-9438777\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\left (317982 i \sqrt {3}+29902 i \sqrt {339}+61863 \sqrt {113}+657999\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-20082360 i \sqrt {3}-1889144 i \sqrt {339}+6112920 \sqrt {113}+64982232\right ) {\mathrm e}^{\frac {t \left (\left (i \sqrt {3}-1\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-16 i \sqrt {3}-4 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}-16\right )}{12 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}}-591894 \left (\left (\sqrt {113}+\frac {9281}{873}\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\left (-\frac {57407 \sqrt {113}}{197298}-\frac {5399}{1746}\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}+\frac {1248448 \sqrt {113}}{98649}+\frac {117440}{873}\right ) {\mathrm e}^{-\frac {\left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}}{2}+\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}-8\right ) t}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}}}{339 \left (\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}-4\right )^{2} \left (873 \sqrt {113}+9281\right )} \\ \end{align*}
Verification of solutions
\[ x = \frac {\left (\left (2466225 i \sqrt {3}+232025 i \sqrt {339}-9438777-887841 \sqrt {113}\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\left (-317982 i \sqrt {3}-29902 i \sqrt {339}+61863 \sqrt {113}+657999\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}+20082360 i \sqrt {3}+1889144 i \sqrt {339}+6112920 \sqrt {113}+64982232\right ) {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-16 i \sqrt {3}+\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}+4 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+16\right ) t}{12 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}}+\left (\left (-2466225 i \sqrt {3}-232025 i \sqrt {339}-887841 \sqrt {113}-9438777\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\left (317982 i \sqrt {3}+29902 i \sqrt {339}+61863 \sqrt {113}+657999\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-20082360 i \sqrt {3}-1889144 i \sqrt {339}+6112920 \sqrt {113}+64982232\right ) {\mathrm e}^{\frac {t \left (\left (i \sqrt {3}-1\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-16 i \sqrt {3}-4 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}-16\right )}{12 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}}-591894 \left (\left (\sqrt {113}+\frac {9281}{873}\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\left (-\frac {57407 \sqrt {113}}{197298}-\frac {5399}{1746}\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}+\frac {1248448 \sqrt {113}}{98649}+\frac {117440}{873}\right ) {\mathrm e}^{-\frac {\left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}}{2}+\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}-8\right ) t}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}}}{339 \left (\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}-4\right )^{2} \left (873 \sqrt {113}+9281\right )} \] Verified OK.
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: 0.375 (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)
\[ x \left (t \right ) = \frac {\left (\left (\left (32 \sqrt {113}+352\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\left (-\sqrt {113}-25\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}+776 \sqrt {113}+8136\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-16\right ) t}{12 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )+32 \sin \left (\frac {\sqrt {3}\, \left (\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-16\right ) t}{12 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right ) \left (\left (\sqrt {113}+11\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\frac {\left (388+36 \sqrt {113}\right )^{\frac {2}{3}} \left (\sqrt {113}+25\right )}{32}\right ) \sqrt {3}\right ) {\mathrm e}^{-\frac {t \left (4+\frac {\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}}{4}+\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}\right )}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}}-32 \left (\left (\sqrt {113}+11\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\left (-\frac {\sqrt {113}}{32}-\frac {25}{32}\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-\frac {97 \sqrt {113}}{8}-\frac {1017}{8}\right ) {\mathrm e}^{-\frac {\left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}}{2}+\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}-8\right ) t}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}}}{1164 \sqrt {113}+12204} \]
✓ 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 )} \]