Internal problem ID [7204]
Internal file name [OUTPUT/6190_Sunday_June_05_2022_04_27_25_PM_95321338/index.tex]
Book: Own collection of miscellaneous problems
Section: section 3.0
Problem number: 14.
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, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime \prime }+y^{\prime }+y=x} \] With initial conditions \begin {align*} [y^{\prime }\left (0\right ) = 0, y \left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 1] \end {align*}
This is higher order nonhomogeneous ODE. Let the solution be \[ y = y_h + y_p \] Where \(y_h\) is the solution to the homogeneous ODE And \(y_p\) is a particular solution to the nonhomogeneous ODE. \(y_h\) is the solution to \[ y^{\prime \prime \prime }+y^{\prime }+y = 0 \] The characteristic equation is \[ \lambda ^{3}+\lambda +1 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}\\ \lambda _2 &= \frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= \frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) x} c_{1} +{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{2} +{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) x} \\ y_2 &= {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} \\ y_3 &= {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }+y^{\prime }+y = x \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ x \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{1, x\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \left \{{\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) x}, {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x}, {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x}\right \} \] Since there is no duplication between the basis function in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis in the UC_set. \[ y_p = A_{2} x +A_{1} \] The unknowns \(\{A_{1}, A_{2}\}\) are found by substituting the above trial solution \(y_p\) into the ODE and comparing coefficients. Substituting the trial solution into the ODE and simplifying gives \[ A_{2} x +A_{1}+A_{2} = x \] Solving for the unknowns by comparing coefficients results in \[ [A_{1} = -1, A_{2} = 1] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = x -1 \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) x} c_{1} +{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{2} +{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{3}\right ) + \left (x -1\right ) \\ \end{align*} Which simplifies to \[ y = {\mathrm e}^{-\frac {\left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) x}{6 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}} c_{1} +{\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) x}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}} c_{2} +{\mathrm e}^{-\frac {x \left (\left (i \sqrt {3}-1\right ) \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+12 i \sqrt {3}+12\right )}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}} c_{3} +x -1 \] Initial conditions are used to solve for the constants of integration.
Looking at the above solution \begin {align*} y = {\mathrm e}^{-\frac {\left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) x}{6 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}} c_{1} +{\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) x}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}} c_{2} +{\mathrm e}^{-\frac {x \left (\left (i \sqrt {3}-1\right ) \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+12 i \sqrt {3}+12\right )}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}} c_{3} +x -1 \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 \(y = 0\) and \(x = 0\) in the above gives \begin {align*} 0 = c_{1} +c_{2} +c_{3} -1\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} y^{\prime } = -\frac {\left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) {\mathrm e}^{-\frac {\left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) x}{6 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}} c_{1}}{6 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}+\frac {\left (i \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) x}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}} c_{2}}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}-\frac {\left (\left (i \sqrt {3}-1\right ) \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+12 i \sqrt {3}+12\right ) {\mathrm e}^{-\frac {x \left (\left (i \sqrt {3}-1\right ) \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+12 i \sqrt {3}+12\right )}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}} c_{3}}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}+1 \end {align*}
substituting \(y^{\prime } = 0\) and \(x = 0\) in the above gives \begin {align*} 0 = \frac {\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}} \left (i \left (c_{2} -c_{3} \right ) \sqrt {3}-2 c_{1} +c_{2} +c_{3} \right )+12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}+12 i \left (c_{2} -c_{3} \right ) \sqrt {3}+24 c_{1} -12 c_{2} -12 c_{3}}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}\tag {2A} \end {align*}
Taking two derivatives of the solution gives \begin {align*} y^{\prime \prime } = \frac {\left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right )^{2} {\mathrm e}^{-\frac {\left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) x}{6 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}} c_{1}}{36 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}}+\frac {\left (i \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right )^{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) x}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}} c_{2}}{144 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}}+\frac {\left (\left (i \sqrt {3}-1\right ) \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+12 i \sqrt {3}+12\right )^{2} {\mathrm e}^{-\frac {x \left (\left (i \sqrt {3}-1\right ) \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+12 i \sqrt {3}+12\right )}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}} c_{3}}{144 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}} \end {align*}
substituting \(y^{\prime \prime } = 1\) and \(x = 0\) in the above gives \begin {align*} 1 = \frac {\frac {3 \left (\left (\frac {\left (2 c_{1} -c_{2} -c_{3} \right ) \sqrt {31}}{9}+i c_{2} -i c_{3} \right ) \sqrt {3}+\frac {i \left (c_{2} -c_{3} \right ) \sqrt {31}}{3}+2 c_{1} -c_{2} -c_{3} \right ) \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}{2}+\frac {2 \left (-c_{1} -c_{2} -c_{3} \right ) \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}}{3}+2 i \left (-c_{2} +c_{3} \right ) \sqrt {3}+4 c_{1} -2 c_{2} -2 c_{3}}{\left (108+12 \sqrt {3}\, \sqrt {31}\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 {\frac {124 \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}}{3}+12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}} \sqrt {31}+124 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}} \sqrt {3}+40 \sqrt {31}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}+1240 \sqrt {3}+384 \sqrt {31}}{\left (31 \sqrt {3}+9 \sqrt {31}\right ) \left (\sqrt {3}\, \sqrt {31}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}-\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+9 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}+12\right )}\\ c_{2}&=\frac {342 i \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}+i \left (108+12 \sqrt {93}\right )^{\frac {2}{3}} \sqrt {93}+38 i \left (108+12 \sqrt {93}\right )^{\frac {1}{3}} \sqrt {93}-54 \sqrt {31}\, \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}-186 \left (108+12 \sqrt {93}\right )^{\frac {1}{3}} \sqrt {3}-300 i \sqrt {93}-7 i \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}+31 \sqrt {3}\, \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}+9 \left (108+12 \sqrt {93}\right )^{\frac {2}{3}} \sqrt {31}-2892 i-372 \sqrt {3}-108 \sqrt {31}}{\left (i \sqrt {3}\, \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}+12 i \sqrt {3}+3 \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}-36\right ) \left (31 \sqrt {3}+9 \sqrt {31}\right )}\\ c_{3}&=-\frac {30 \sqrt {31}\, \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}+19 \sqrt {3}\, \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}+3 \left (108+12 \sqrt {93}\right )^{\frac {2}{3}} \sqrt {31}+3 i \left (108+12 \sqrt {93}\right )^{\frac {2}{3}} \sqrt {93}+78 \left (108+12 \sqrt {93}\right )^{\frac {1}{3}} \sqrt {3}-30 i \left (108+12 \sqrt {93}\right )^{\frac {1}{3}} \sqrt {93}-744 \sqrt {3}-216 \sqrt {31}+57 i \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}-234 i \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}}{72 \left (31 \sqrt {3}+9 \sqrt {31}\right )} \end {align*}
Substituting these values back in above solution results in \begin {align*} y = \text {Expression too large to display} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (\left (-35238 \sqrt {31}-113274 \sqrt {3}+44922 i+4658 i \sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (-3670 i \sqrt {93}-35392 i+6076 \sqrt {3}+1890 \sqrt {31}\right ) \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}+91872 i \sqrt {93}+885984 i+388368 \sqrt {3}+120816 \sqrt {31}\right ) {\mathrm e}^{\frac {\left (-i \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12 i \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) x}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}}+\left (\left (-37572 \sqrt {3}-11688 \sqrt {31}-120624 i-12508 i \sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (2381 i \sqrt {93}+22961 i-6355 \sqrt {3}-1977 \sqrt {31}\right ) \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}-27144 i \sqrt {93}-261768 i+388368 \sqrt {3}+120816 \sqrt {31}\right ) {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) x}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}}+\left (\left (16722 \sqrt {31}+53754 \sqrt {3}+75702 i+7850 i \sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (1289 i \sqrt {93}+12431 i+5487 \sqrt {3}+1707 \sqrt {31}\right ) \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}+32364 i \sqrt {93}+312108 i+97092 \sqrt {3}+30204 \sqrt {31}\right ) {\mathrm e}^{-\frac {\left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) x}{6 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}}+97092 \left (x -1\right ) \left (\left (-\sqrt {3}-\frac {839 \sqrt {31}}{2697}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (\frac {14 \sqrt {3}}{261}+\frac {15 \sqrt {31}}{899}\right ) \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}+i \sqrt {93}+\frac {839 i}{87}+9 \sqrt {3}+\frac {2517 \sqrt {31}}{899}\right )}{\left (-97092 \sqrt {3}-30204 \sqrt {31}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (5208 \sqrt {3}+1620 \sqrt {31}\right ) \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}+97092 i \sqrt {93}+936324 i+873828 \sqrt {3}+271836 \sqrt {31}} \\ \end{align*}
Verification of solutions
\[ y = \frac {\left (\left (-35238 \sqrt {31}-113274 \sqrt {3}+44922 i+4658 i \sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (-3670 i \sqrt {93}-35392 i+6076 \sqrt {3}+1890 \sqrt {31}\right ) \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}+91872 i \sqrt {93}+885984 i+388368 \sqrt {3}+120816 \sqrt {31}\right ) {\mathrm e}^{\frac {\left (-i \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12 i \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) x}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}}+\left (\left (-37572 \sqrt {3}-11688 \sqrt {31}-120624 i-12508 i \sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (2381 i \sqrt {93}+22961 i-6355 \sqrt {3}-1977 \sqrt {31}\right ) \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}-27144 i \sqrt {93}-261768 i+388368 \sqrt {3}+120816 \sqrt {31}\right ) {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) x}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}}+\left (\left (16722 \sqrt {31}+53754 \sqrt {3}+75702 i+7850 i \sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (1289 i \sqrt {93}+12431 i+5487 \sqrt {3}+1707 \sqrt {31}\right ) \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}+32364 i \sqrt {93}+312108 i+97092 \sqrt {3}+30204 \sqrt {31}\right ) {\mathrm e}^{-\frac {\left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) x}{6 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}}+97092 \left (x -1\right ) \left (\left (-\sqrt {3}-\frac {839 \sqrt {31}}{2697}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (\frac {14 \sqrt {3}}{261}+\frac {15 \sqrt {31}}{899}\right ) \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}+i \sqrt {93}+\frac {839 i}{87}+9 \sqrt {3}+\frac {2517 \sqrt {31}}{899}\right )}{\left (-97092 \sqrt {3}-30204 \sqrt {31}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (5208 \sqrt {3}+1620 \sqrt {31}\right ) \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}+97092 i \sqrt {93}+936324 i+873828 \sqrt {3}+271836 \sqrt {31}} \] Verified OK.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 3; linear nonhomogeneous with symmetry [0,1] trying high order linear exact nonhomogeneous trying differential order: 3; missing the dependent variable checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.609 (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 \left (x \right ) = \frac {\frac {10 \,{\mathrm e}^{-\frac {x \left (108+12 \sqrt {93}\right )^{\frac {1}{3}} \left (-12+\left (\sqrt {93}-9\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}\right )}{144}} \left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {31}+\frac {3 \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}} \sqrt {31}}{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 )}{3}-26 \,{\mathrm e}^{-\frac {x \left (108+12 \sqrt {93}\right )^{\frac {1}{3}} \left (-12+\left (\sqrt {93}-9\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}\right )}{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 )+\left (-76-\frac {10 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {31}}{3}+\sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}} \sqrt {31}+4 \sqrt {3}\, \sqrt {31}+26 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}-\frac {31 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}}{3}\right ) {\mathrm e}^{\frac {x \left (108+12 \sqrt {93}\right )^{\frac {1}{3}} \left (-12+\left (\sqrt {93}-9\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}\right )}{72}}+3 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}} \left (x -1\right ) \left (\sqrt {3}\, \sqrt {31}-\frac {31}{3}\right )}{\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}} \left (3 \sqrt {3}\, \sqrt {31}-31\right )} \]
✓ Solution by Mathematica
Time used: 0.024 (sec). Leaf size: 1546
DSolve[{y'''[x]+y'[x]+y[x]==x,{y'[1] == 0,y[0]==0,y''[0]==1}},y[x],x,IncludeSingularSolutions -> True]
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