Internal problem ID [6451]
Book: Own collection of miscellaneous problems
Section: section 3.0
Problem number: 14.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime }+y^{\prime }+y-x=0} \end {gather*} With initial conditions \begin {align*} [y^{\prime }\relax (0) = 0, y \relax (0) = 0, y^{\prime \prime }\relax (0) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.338 (sec). Leaf size: 359
dsolve([diff(y(x),x$3)+diff(y(x),x)+y(x)=x,D(y)(0) = 0, y(0) = 0, (D@@2)(y)(0) = 1],y(x), singsol=all)
\[ y \relax (x ) = \frac {10 \,{\mathrm e}^{-\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}} \left (-12+\left (-9+\sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}\right ) x}{144}} \left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {31}+\frac {3 \sqrt {31}\, \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}}{5}-\frac {6 \sqrt {3}\, \sqrt {31}}{5}-\frac {39 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}{5}-\frac {31 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}}{5}+\frac {114}{5}\right ) \cos \left (\frac {\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}} \sqrt {3}\, \left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {31}-9 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}+12\right ) x}{144}\right )-78 \,{\mathrm e}^{-\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}} \left (-12+\left (-9+\sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}\right ) x}{144}} \left (\left (\sqrt {3}-\frac {5 \sqrt {31}}{13}\right ) \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}+\frac {38 \sqrt {3}}{13}-\frac {6 \sqrt {31}}{13}\right ) \sin \left (\frac {\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}} \sqrt {3}\, \left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {31}-9 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}+12\right ) x}{144}\right )+9 \left (-\frac {10 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {31}}{9}+\frac {\sqrt {31}\, \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}}{3}+\frac {4 \sqrt {3}\, \sqrt {31}}{3}+\frac {26 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}{3}-\frac {31 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}}{9}-\frac {76}{3}\right ) {\mathrm e}^{\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}} \left (-12+\left (-9+\sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}\right ) x}{72}}+9 \left (x -1\right ) \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}} \left (\sqrt {3}\, \sqrt {31}-\frac {31}{3}\right )}{\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}} \left (9 \sqrt {3}\, \sqrt {31}-93\right )} \]
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 731
DSolve[{y'''[x]+y'[x]+y[x]==x,{y'[1] == 0,y[0]==0,y''[0]==1}},y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\text {Root}\left [\text {$\#$1}^6+3 \text {$\#$1}^5+11 \text {$\#$1}^4+23 \text {$\#$1}^3+17 \text {$\#$1}^2+3 \text {$\#$1}+9\&,5\right ] \exp \left (x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,3\right ]+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,2\right ]\right )+(x-1) \text {Root}\left [\text {$\#$1}^6+11 \text {$\#$1}^4+38 \text {$\#$1}^2+31\&,2\right ] \exp \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,3\right ]\right )+\text {Root}\left [\text {$\#$1}^6+3 \text {$\#$1}^5+11 \text {$\#$1}^4+23 \text {$\#$1}^3+17 \text {$\#$1}^2+3 \text {$\#$1}+9\&,2\right ] \exp \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,1\right ]+x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,2\right ]\right )+\text {Root}\left [\text {$\#$1}^6-3 \text {$\#$1}^5+11 \text {$\#$1}^4-23 \text {$\#$1}^3+17 \text {$\#$1}^2-3 \text {$\#$1}+9\&,6\right ] \exp \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,1\right ]+x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,3\right ]\right )+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,2\right ] \left (-1+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,3\right ]^2\right ) \exp \left (x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,1\right ]+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,2\right ]\right )-\left (-1+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,2\right ]^2\right ) \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,3\right ] \exp \left (x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,1\right ]+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,3\right ]\right )+\left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,1\right ]^2-1\right ) \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,3\right ] \exp \left (x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,3\right ]\right )+\text {Root}\left [\text {$\#$1}^6-2 \text {$\#$1}^4+\text {$\#$1}^2+31\&,1\right ] \exp \left (x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,3\right ]\right )+\text {Root}\left [\text {$\#$1}^6-2 \text {$\#$1}^4+\text {$\#$1}^2+31\&,5\right ] \exp \left (x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,2\right ]\right )+(x-1) \text {Root}\left [\text {$\#$1}^6+11 \text {$\#$1}^4+38 \text {$\#$1}^2+31\&,6\right ] \exp \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,2\right ]\right )+(x-1) \text {Root}\left [\text {$\#$1}^6+11 \text {$\#$1}^4+38 \text {$\#$1}^2+31\&,4\right ] \exp \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,1\right ]\right )+\text {Root}\left [\text {$\#$1}^6-2 \text {$\#$1}^4+\text {$\#$1}^2+31\&,4\right ] \exp \left (x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,1\right ]\right )}{\text {Root}\left [\text {$\#$1}^6+11 \text {$\#$1}^4+38 \text {$\#$1}^2+31\&,2\right ] \exp \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,3\right ]\right )+\text {Root}\left [\text {$\#$1}^6+11 \text {$\#$1}^4+38 \text {$\#$1}^2+31\&,6\right ] \exp \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,2\right ]\right )+\text {Root}\left [\text {$\#$1}^6+11 \text {$\#$1}^4+38 \text {$\#$1}^2+31\&,4\right ] \exp \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,1\right ]\right )} \\ \end{align*}