Internal problem ID [11000]
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.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime \prime }-5 y^{\prime \prime }+y^{\prime }-y+t^{2}-2 t +10=0} \] 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*}
✓ Solution by Maple
Time used: 0.453 (sec). Leaf size: 369
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 {154 \left (\left (116+6 \sqrt {3}\, \sqrt {26}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {26}+\frac {58 \left (116+6 \sqrt {3}\, \sqrt {26}\right )^{\frac {2}{3}} \sqrt {26}\, \sqrt {3}}{77}+\frac {55 \sqrt {3}\, \sqrt {26}}{14}-\frac {69 \left (116+6 \sqrt {3}\, \sqrt {26}\right )^{\frac {1}{3}}}{14}-\frac {234 \left (116+6 \sqrt {3}\, \sqrt {26}\right )^{\frac {2}{3}}}{77}+\frac {33}{7}\right ) {\mathrm e}^{\frac {\left (3 \sqrt {78}\, \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}-58 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}-242 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}+2420\right ) t}{1452}} \cos \left (\frac {\left (116+6 \sqrt {3}\, \sqrt {26}\right )^{\frac {1}{3}} \sqrt {3}\, \left (3 \left (116+6 \sqrt {3}\, \sqrt {26}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {26}-58 \left (116+6 \sqrt {3}\, \sqrt {26}\right )^{\frac {1}{3}}+242\right ) t}{1452}\right )+759 \,{\mathrm e}^{\frac {\left (3 \sqrt {78}\, \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}-58 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}-242 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}+2420\right ) t}{1452}} \left (\left (\sqrt {3}-\frac {14 \sqrt {26}}{23}\right ) \left (116+6 \sqrt {3}\, \sqrt {26}\right )^{\frac {1}{3}}+\frac {22 \sqrt {3}}{23}+\frac {55 \sqrt {26}}{23}\right ) \sin \left (\frac {\left (116+6 \sqrt {3}\, \sqrt {26}\right )^{\frac {1}{3}} \sqrt {3}\, \left (3 \left (116+6 \sqrt {3}\, \sqrt {26}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {26}-58 \left (116+6 \sqrt {3}\, \sqrt {26}\right )^{\frac {1}{3}}+242\right ) t}{1452}\right )+87 \left (-\frac {154 \left (116+6 \sqrt {3}\, \sqrt {26}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {26}}{87}+\frac {2 \left (116+6 \sqrt {3}\, \sqrt {26}\right )^{\frac {2}{3}} \sqrt {26}\, \sqrt {3}}{3}-\frac {605 \sqrt {3}\, \sqrt {26}}{87}+\frac {253 \left (116+6 \sqrt {3}\, \sqrt {26}\right )^{\frac {1}{3}}}{29}-\frac {78 \left (116+6 \sqrt {3}\, \sqrt {26}\right )^{\frac {2}{3}}}{29}-\frac {242}{29}\right ) {\mathrm e}^{-\frac {\left (3 \sqrt {78}\, \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}-58 \left (116+6 \sqrt {78}\right )^{\frac {2}{3}}-242 \left (116+6 \sqrt {78}\right )^{\frac {1}{3}}-1210\right ) t}{726}}+87 t^{2} \left (116+6 \sqrt {3}\, \sqrt {26}\right )^{\frac {2}{3}} \left (\sqrt {3}\, \sqrt {26}-\frac {117}{29}\right )}{\left (116+6 \sqrt {3}\, \sqrt {26}\right )^{\frac {2}{3}} \left (87 \sqrt {3}\, \sqrt {26}-351\right )} \]
✓ Solution by Mathematica
Time used: 0.01 (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]
\begin{align*} 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 )} \\ \end{align*}