Internal problem ID [1571]
Internal file name [OUTPUT/1572_Sunday_June_05_2022_02_22_59_AM_20540918/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 9 Introduction to Linear Higher Order Equations. Section 9.3. Undetermined
Coefficients for Higher Order Equations. Page 495
Problem number: section 9.3, problem 74.
ODE order: 4.
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
[[_high_order, _missing_y]]
\[ \boxed {y^{\prime \prime \prime \prime }-3 y^{\prime \prime \prime }+5 y^{\prime \prime }-2 y^{\prime }=-2 \,{\mathrm e}^{x} \left (\cos \left (x \right )-\sin \left (x \right )\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = -1, y^{\prime \prime \prime }\left (0\right ) = -5] \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 \[ y^{\prime \prime \prime \prime }-3 y^{\prime \prime \prime }+5 y^{\prime \prime }-2 y^{\prime } = 0 \] The characteristic equation is \[ \lambda ^{4}-3 \lambda ^{3}+5 \lambda ^{2}-2 \lambda = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= -\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}+\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1\\ \lambda _3 &= \frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _4 &= \frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} +{\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}+\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1\right ) x} c_{2} +{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{3} +{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= {\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}+\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1\right ) x} \\ y_3 &= {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} \\ y_4 &= {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }-3 y^{\prime \prime \prime }+5 y^{\prime \prime }-2 y^{\prime } = -2 \,{\mathrm e}^{x} \left (\cos \left (x \right )-\sin \left (x \right )\right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ -2 \,{\mathrm e}^{x} \left (\cos \left (x \right )-\sin \left (x \right )\right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{\cos \left (x \right ) {\mathrm e}^{x}, \sin \left (x \right ) {\mathrm e}^{x}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \left \{1, {\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}+\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1\right ) x}, {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x}, {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x}\right \} \] 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. \[ y_p = A_{1} \cos \left (x \right ) {\mathrm e}^{x}+A_{2} \sin \left (x \right ) {\mathrm e}^{x} \] 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 \[ -2 A_{1} \sin \left (x \right ) {\mathrm e}^{x}+2 A_{2} \cos \left (x \right ) {\mathrm e}^{x} = -2 \,{\mathrm e}^{x} \left (\cos \left (x \right )-\sin \left (x \right )\right ) \] 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 \[ y_p = -\cos \left (x \right ) {\mathrm e}^{x}-\sin \left (x \right ) {\mathrm e}^{x} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} +{\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}+\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1\right ) x} c_{2} +{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{3} +{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{4}\right ) + \left (-\cos \left (x \right ) {\mathrm e}^{x}-\sin \left (x \right ) {\mathrm e}^{x}\right ) \\ \end{align*} Initial conditions are used to solve for the constants of integration.
Looking at the above solution \begin {align*} y = c_{1} +{\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}+\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1\right ) x} c_{2} +{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{3} +{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{4} -\cos \left (x \right ) {\mathrm e}^{x}-\sin \left (x \right ) {\mathrm e}^{x} \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 \(y = 2\) and \(x = 0\) in the above gives \begin {align*} 2 = -1+c_{1} +c_{2} +c_{4} +c_{3}\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} y^{\prime } = \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}+\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1\right ) {\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}+\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1\right ) x} c_{2} +\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{3} +\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{4} -2 \cos \left (x \right ) {\mathrm e}^{x} \end {align*}
substituting \(y^{\prime } = 0\) and \(x = 0\) in the above gives \begin {align*} 0 = \frac {\left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}} \left (i \left (c_{3} -c_{4} \right ) \sqrt {3}-2 c_{2} +c_{3} +c_{4} \right )+12 \left (-2+c_{2} +c_{3} +c_{4} \right ) \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}+24 i \left (c_{3} -c_{4} \right ) \sqrt {3}+48 c_{2} -24 c_{3} -24 c_{4}}{12 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}}\tag {2A} \end {align*}
Taking two derivatives of the solution gives \begin {align*} y^{\prime \prime } = \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}+\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1\right )^{2} {\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}+\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1\right ) x} c_{2} +\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2} {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{3} +\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2} {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{4} +2 \sin \left (x \right ) {\mathrm e}^{x}-2 \cos \left (x \right ) {\mathrm e}^{x} \end {align*}
substituting \(y^{\prime \prime } = -1\) and \(x = 0\) in the above gives \begin {align*} -1 = \frac {\frac {11 \left (\frac {\left (\frac {\left (2 c_{2} -c_{3} -c_{4} \right ) \sqrt {3}}{3}+i c_{3} -i c_{4} \right ) \sqrt {59}}{11}+i \left (c_{3} -c_{4} \right ) \sqrt {3}+2 c_{2} -c_{3} -c_{4} \right ) \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}}{2}+\left (-2-\frac {c_{2}}{3}-\frac {c_{3}}{3}-\frac {c_{4}}{3}\right ) \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}+2 \left (\left (-2 c_{2} +c_{3} +c_{4} \right ) \sqrt {3}+3 i c_{3} -3 i c_{4} \right ) \sqrt {59}+10 i \left (c_{3} -c_{4} \right ) \sqrt {3}-20 c_{2} +10 c_{3} +10 c_{4}}{\left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}}\tag {3A} \end {align*}
Taking three derivatives of the solution gives \begin {align*} y^{\prime \prime \prime } = \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}+\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1\right )^{3} {\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}+\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1\right ) x} c_{2} +\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right )^{3} {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{3} +\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right )^{3} {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{4} +4 \sin \left (x \right ) {\mathrm e}^{x} \end {align*}
substituting \(y^{\prime \prime \prime } = -5\) and \(x = 0\) in the above gives \begin {align*} -5 = -\frac {2 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}} \left (\left (-\left (-2 c_{2} +c_{3} +c_{4} \right ) \sqrt {3}-3 i c_{3} +3 i c_{4} \right ) \sqrt {59}-15 i \left (c_{4} -c_{3} \right ) \sqrt {3}-30 c_{2} +15 c_{3} +15 c_{4} \right )+\left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}} \left (\left (-\left (2 c_{2} -c_{3} -c_{4} \right ) \sqrt {3}-3 i c_{3} +3 i c_{4} \right ) \sqrt {59}-13 i \left (c_{3} -c_{4} \right ) \sqrt {3}-26 c_{2} +13 c_{3} +13 c_{4} \right )+96 \left (\sqrt {3}\, \sqrt {59}+9\right ) \left (c_{2} +c_{3} +c_{4} \right )}{24 \left (\sqrt {3}\, \sqrt {59}+9\right )}\tag {4A} \end {align*}
Equations {1A,2A,3A,4A} are now solved for \(\{c_{1}, c_{2}, c_{3}, c_{4}\}\). Solving for the constants gives \begin {align*} c_{1}&=2\\ c_{2}&=\frac {11895552 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}} \sqrt {59}+52607232 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}} \sqrt {3}-361827648 \sqrt {59}\, \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}-1602583488 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}} \sqrt {3}+1828407168 \sqrt {59}+8118029952 \sqrt {3}}{\left (\left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}-6 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}-24\right )^{2} \left (177 \sqrt {3}+43 \sqrt {59}\right ) \left (96 \sqrt {59}\, \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}+\sqrt {59}\, \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}-540 \sqrt {3}\, \sqrt {59}+1248 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}+57 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}-7740\right )}\\ c_{3}&=\frac {10 i \sqrt {59}\, \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}-153 i \sqrt {59}\, \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}+1050 i \sqrt {3}\, \sqrt {59}+97 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}} \sqrt {3}+90 i \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}+27 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}} \sqrt {59}-531 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}} \sqrt {3}-2193 i \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}-129 \sqrt {59}\, \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}-7230 \sqrt {3}+13770 i-1674 \sqrt {59}}{3 \left (\left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}-6 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}-24\right ) \left (177 \sqrt {3}+43 \sqrt {59}\right )}\\ c_{4}&=\frac {-153648 \left (108+12 \sqrt {177}\right )^{\frac {2}{3}} \sqrt {177}+226800 \left (108+12 \sqrt {177}\right )^{\frac {1}{3}} \sqrt {177}+4084128 \sqrt {177}+61920 i \sqrt {59}\, \left (108+12 \sqrt {177}\right )^{\frac {2}{3}}-2011824 \left (108+12 \sqrt {177}\right )^{\frac {2}{3}}+254880 i \left (108+12 \sqrt {177}\right )^{\frac {2}{3}} \sqrt {3}-1520208 i \sqrt {59}\, \left (108+12 \sqrt {177}\right )^{\frac {1}{3}}+2974320 \left (108+12 \sqrt {177}\right )^{\frac {1}{3}}-6839280 i \sqrt {3}\, \left (108+12 \sqrt {177}\right )^{\frac {1}{3}}+7434720 i \sqrt {59}+53636256+32879520 i \sqrt {3}}{\left (\left (108+12 \sqrt {177}\right )^{\frac {2}{3}}-6 \left (108+12 \sqrt {177}\right )^{\frac {1}{3}}-24\right ) \left (177 \sqrt {3}+43 \sqrt {59}\right ) \left (42 \sqrt {59}\, \left (108+12 \sqrt {177}\right )^{\frac {1}{3}}+37 \left (108+12 \sqrt {177}\right )^{\frac {2}{3}} \sqrt {3}+3 \sqrt {59}\, \left (108+12 \sqrt {177}\right )^{\frac {2}{3}}-3 i \left (108+12 \sqrt {177}\right )^{\frac {2}{3}} \sqrt {177}+78 \sqrt {3}\, \left (108+12 \sqrt {177}\right )^{\frac {1}{3}}+42 i \left (108+12 \sqrt {177}\right )^{\frac {1}{3}} \sqrt {177}-1416 \sqrt {3}-216 \sqrt {59}-111 i \left (108+12 \sqrt {177}\right )^{\frac {2}{3}}+234 i \left (108+12 \sqrt {177}\right )^{\frac {1}{3}}\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} \text {Expression too large to display} \\ \end{align*}
Verification of solutions
\[ \text {Expression too large to display} \] Verified OK.
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 4; linear nonhomogeneous with symmetry [0,1] -> Calling odsolve with the ODE`, diff(diff(diff(_b(_a), _a), _a), _a) = -2*exp(_a)*cos(_a)+2*exp(_a)*sin(_a)+3*(diff(diff(_b(_a), _ 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 <- differential order: 4; linear nonhomogeneous with symmetry [0,1] successful`
✓ Solution by Maple
Time used: 4.203 (sec). Leaf size: 1300
dsolve([1*diff(y(x),x$4)-3*diff(y(x),x$3)+5*diff(y(x),x$2)-2*diff(y(x),x)+0*y(x)=-2*exp(x)*(cos(x)-sin(x)),y(0) = 2, D(y)(0) = 0, (D@@2)(y)(0) = -1, (D@@3)(y)(0) = -5],y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 0.103 (sec). Leaf size: 3484
DSolve[{1*y''''[x]-3*y'''[x]+5*y''[x]-2*y'[x]+0*y[x]==-2*Exp[x]*(Cos[x]-Sin[x]),{y[0]==2,y'[0]==0,y''[0]==-1,y'''[0]==-5}},y[x],x,IncludeSingularSolutions -> True]
Too large to display