Internal problem ID [1526]
Internal file name [OUTPUT/1527_Sunday_June_05_2022_02_20_43_AM_72947374/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 29.
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, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime \prime }-y^{\prime \prime }-4 y^{\prime }+4 y={\mathrm e}^{-x} \left (\left (16+10 x \right ) \cos \left (x \right )+\left (30-10 x \right ) \sin \left (x \right )\right )} \] 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 \[ y^{\prime \prime \prime }-y^{\prime \prime }-4 y^{\prime }+4 y = 0 \] The characteristic equation is \[ \lambda ^{3}-\lambda ^{2}-4 \lambda +4 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 1\\ \lambda _2 &= 2\\ \lambda _3 &= -2 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{x}+c_{3} {\mathrm e}^{2 x} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{-2 x} \\ y_2 &= {\mathrm e}^{x} \\ y_3 &= {\mathrm e}^{2 x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }-y^{\prime \prime }-4 y^{\prime }+4 y = {\mathrm e}^{-x} \left (\left (16+10 x \right ) \cos \left (x \right )+\left (30-10 x \right ) \sin \left (x \right )\right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ {\mathrm e}^{-x} \left (\left (16+10 x \right ) \cos \left (x \right )+\left (30-10 x \right ) \sin \left (x \right )\right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{\cos \left (x \right ) {\mathrm e}^{-x}, \sin \left (x \right ) {\mathrm e}^{-x}, x \cos \left (x \right ) {\mathrm e}^{-x}, \sin \left (x \right ) x \,{\mathrm e}^{-x}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{{\mathrm e}^{x}, {\mathrm e}^{-2 x}, {\mathrm e}^{2 x}\} \] 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. \[ y_p = A_{1} \cos \left (x \right ) {\mathrm e}^{-x}+A_{2} \sin \left (x \right ) {\mathrm e}^{-x}+A_{3} x \cos \left (x \right ) {\mathrm e}^{-x}+A_{4} \sin \left (x \right ) x \,{\mathrm e}^{-x} \] The unknowns \(\{A_{1}, A_{2}, A_{3}, A_{4}\}\) 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 \[ 10 A_{1} \cos \left (x \right ) {\mathrm e}^{-x}+10 A_{2} \sin \left (x \right ) {\mathrm e}^{-x}-2 A_{4} \sin \left (x \right ) {\mathrm e}^{-x}+10 A_{4} \sin \left (x \right ) x \,{\mathrm e}^{-x}-2 A_{3} \cos \left (x \right ) {\mathrm e}^{-x}+10 A_{3} x \cos \left (x \right ) {\mathrm e}^{-x}-8 A_{4} \cos \left (x \right ) {\mathrm e}^{-x}+8 A_{3} \sin \left (x \right ) {\mathrm e}^{-x} = {\mathrm e}^{-x} \left (\left (16+10 x \right ) \cos \left (x \right )+\left (30-10 x \right ) \sin \left (x \right )\right ) \] Solving for the unknowns by comparing coefficients results in \[ [A_{1} = 1, A_{2} = 2, A_{3} = 1, A_{4} = -1] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \cos \left (x \right ) {\mathrm e}^{-x}+2 \sin \left (x \right ) {\mathrm e}^{-x}+x \cos \left (x \right ) {\mathrm e}^{-x}-\sin \left (x \right ) x \,{\mathrm e}^{-x} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{x}+c_{3} {\mathrm e}^{2 x}\right ) + \left (\cos \left (x \right ) {\mathrm e}^{-x}+2 \sin \left (x \right ) {\mathrm e}^{-x}+x \cos \left (x \right ) {\mathrm e}^{-x}-\sin \left (x \right ) x \,{\mathrm e}^{-x}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{x}+c_{3} {\mathrm e}^{2 x}+\cos \left (x \right ) {\mathrm e}^{-x}+2 \sin \left (x \right ) {\mathrm e}^{-x}+x \cos \left (x \right ) {\mathrm e}^{-x}-\sin \left (x \right ) x \,{\mathrm e}^{-x} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{x}+c_{3} {\mathrm e}^{2 x}+\cos \left (x \right ) {\mathrm e}^{-x}+2 \sin \left (x \right ) {\mathrm e}^{-x}+x \cos \left (x \right ) {\mathrm e}^{-x}-\sin \left (x \right ) x \,{\mathrm e}^{-x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }-y^{\prime \prime }-4 y^{\prime }+4 y={\mathrm e}^{-x} \left (\left (16+10 x \right ) \cos \left (x \right )+\left (30-10 x \right ) \sin \left (x \right )\right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & y^{\prime \prime \prime } \\ \bullet & {} & \textrm {Isolate 3rd derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }=-4 y-10 \sin \left (x \right ) x \,{\mathrm e}^{-x}+10 x \cos \left (x \right ) {\mathrm e}^{-x}+30 \sin \left (x \right ) {\mathrm e}^{-x}+16 \cos \left (x \right ) {\mathrm e}^{-x}+y^{\prime \prime }+4 y^{\prime } \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }-y^{\prime \prime }-4 y^{\prime }+4 y=-2 \,{\mathrm e}^{-x} \left (-5 x \cos \left (x \right )+5 \sin \left (x \right ) x -8 \cos \left (x \right )-15 \sin \left (x \right )\right ) \\ \square & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{1}\left (x \right ) \\ {} & {} & y_{1}\left (x \right )=y \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{2}\left (x \right ) \\ {} & {} & y_{2}\left (x \right )=y^{\prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{3}\left (x \right ) \\ {} & {} & y_{3}\left (x \right )=y^{\prime \prime } \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} y_{3}^{\prime }\left (x \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{3}^{\prime }\left (x \right )=-10 \sin \left (x \right ) x \,{\mathrm e}^{-x}+10 x \cos \left (x \right ) {\mathrm e}^{-x}+30 \sin \left (x \right ) {\mathrm e}^{-x}+16 \cos \left (x \right ) {\mathrm e}^{-x}+y_{3}\left (x \right )+4 y_{2}\left (x \right )-4 y_{1}\left (x \right ) \\ & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & {} & \left [y_{2}\left (x \right )=y_{1}^{\prime }\left (x \right ), y_{3}\left (x \right )=y_{2}^{\prime }\left (x \right ), y_{3}^{\prime }\left (x \right )=-10 \sin \left (x \right ) x \,{\mathrm e}^{-x}+10 x \cos \left (x \right ) {\mathrm e}^{-x}+30 \sin \left (x \right ) {\mathrm e}^{-x}+16 \cos \left (x \right ) {\mathrm e}^{-x}+y_{3}\left (x \right )+4 y_{2}\left (x \right )-4 y_{1}\left (x \right )\right ] \\ \bullet & {} & \textrm {Define vector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}\left (x \right )=\left [\begin {array}{c} y_{1}\left (x \right ) \\ y_{2}\left (x \right ) \\ y_{3}\left (x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -4 & 4 & 1 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \right )+\left [\begin {array}{c} 0 \\ 0 \\ 10 x \cos \left (x \right ) {\mathrm e}^{-x}-10 \sin \left (x \right ) x \,{\mathrm e}^{-x}+16 \cos \left (x \right ) {\mathrm e}^{-x}+30 \sin \left (x \right ) {\mathrm e}^{-x} \end {array}\right ] \\ \bullet & {} & \textrm {Define the forcing function}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{f}}\left (x \right )=\left [\begin {array}{c} 0 \\ 0 \\ 10 x \cos \left (x \right ) {\mathrm e}^{-x}-10 \sin \left (x \right ) x \,{\mathrm e}^{-x}+16 \cos \left (x \right ) {\mathrm e}^{-x}+30 \sin \left (x \right ) {\mathrm e}^{-x} \end {array}\right ] \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -4 & 4 & 1 \end {array}\right ] \\ \bullet & {} & \textrm {Rewrite the system as}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=A \cdot {\moverset {\rightarrow }{y}}\left (x \right )+{\moverset {\rightarrow }{f}} \\ \bullet & {} & \textrm {To solve the system, find the eigenvalues and eigenvectors of}\hspace {3pt} A \\ \bullet & {} & \textrm {Eigenpairs of}\hspace {3pt} A \\ {} & {} & \left [\left [-2, \left [\begin {array}{c} \frac {1}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ]\right ], \left [1, \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ]\right ], \left [2, \left [\begin {array}{c} \frac {1}{4} \\ \frac {1}{2} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-2, \left [\begin {array}{c} \frac {1}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{-2 x}\cdot \left [\begin {array}{c} \frac {1}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [1, \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{2}={\mathrm e}^{x}\cdot \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [2, \left [\begin {array}{c} \frac {1}{4} \\ \frac {1}{2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{3}={\mathrm e}^{2 x}\cdot \left [\begin {array}{c} \frac {1}{4} \\ \frac {1}{2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {General solution of the system of ODEs can be written in terms of the particular solution}\hspace {3pt} {\moverset {\rightarrow }{y}}_{p}\left (x \right ) \\ {} & {} & {\moverset {\rightarrow }{y}}\left (x \right )=c_{1} {\moverset {\rightarrow }{y}}_{1}+c_{2} {\moverset {\rightarrow }{y}}_{2}+c_{3} {\moverset {\rightarrow }{y}}_{3}+{\moverset {\rightarrow }{y}}_{p}\left (x \right ) \\ \square & {} & \textrm {Fundamental matrix}\hspace {3pt} \\ {} & \circ & \textrm {Let}\hspace {3pt} \phi \left (x \right )\hspace {3pt}\textrm {be the matrix whose columns are the independent solutions of the homogeneous system.}\hspace {3pt} \\ {} & {} & \phi \left (x \right )=\left [\begin {array}{ccc} \frac {{\mathrm e}^{-2 x}}{4} & {\mathrm e}^{x} & \frac {{\mathrm e}^{2 x}}{4} \\ -\frac {{\mathrm e}^{-2 x}}{2} & {\mathrm e}^{x} & \frac {{\mathrm e}^{2 x}}{2} \\ {\mathrm e}^{-2 x} & {\mathrm e}^{x} & {\mathrm e}^{2 x} \end {array}\right ] \\ {} & \circ & \textrm {The fundamental matrix,}\hspace {3pt} \Phi \left (x \right )\hspace {3pt}\textrm {is a normalized version of}\hspace {3pt} \phi \left (x \right )\hspace {3pt}\textrm {satisfying}\hspace {3pt} \Phi \left (0\right )=I \hspace {3pt}\textrm {where}\hspace {3pt} I \hspace {3pt}\textrm {is the identity matrix}\hspace {3pt} \\ {} & {} & \Phi \left (x \right )=\phi \left (x \right )\cdot \frac {1}{\phi \left (0\right )} \\ {} & \circ & \textrm {Substitute the value of}\hspace {3pt} \phi \left (x \right )\hspace {3pt}\textrm {and}\hspace {3pt} \phi \left (0\right ) \\ {} & {} & \Phi \left (x \right )=\left [\begin {array}{ccc} \frac {{\mathrm e}^{-2 x}}{4} & {\mathrm e}^{x} & \frac {{\mathrm e}^{2 x}}{4} \\ -\frac {{\mathrm e}^{-2 x}}{2} & {\mathrm e}^{x} & \frac {{\mathrm e}^{2 x}}{2} \\ {\mathrm e}^{-2 x} & {\mathrm e}^{x} & {\mathrm e}^{2 x} \end {array}\right ]\cdot \frac {1}{\left [\begin {array}{ccc} \frac {1}{4} & 1 & \frac {1}{4} \\ -\frac {1}{2} & 1 & \frac {1}{2} \\ 1 & 1 & 1 \end {array}\right ]} \\ {} & \circ & \textrm {Evaluate and simplify to get the fundamental matrix}\hspace {3pt} \\ {} & {} & \Phi \left (x \right )=\left [\begin {array}{ccc} -\frac {\left (3 \,{\mathrm e}^{4 x}-8 \,{\mathrm e}^{3 x}-1\right ) {\mathrm e}^{-2 x}}{6} & -\frac {{\mathrm e}^{-2 x}}{4}+\frac {{\mathrm e}^{2 x}}{4} & \frac {\left (3 \,{\mathrm e}^{4 x}-4 \,{\mathrm e}^{3 x}+1\right ) {\mathrm e}^{-2 x}}{12} \\ -\frac {\left (3 \,{\mathrm e}^{4 x}-4 \,{\mathrm e}^{3 x}+1\right ) {\mathrm e}^{-2 x}}{3} & \frac {{\mathrm e}^{-2 x}}{2}+\frac {{\mathrm e}^{2 x}}{2} & \frac {\left (3 \,{\mathrm e}^{4 x}-2 \,{\mathrm e}^{3 x}-1\right ) {\mathrm e}^{-2 x}}{6} \\ -\frac {2 \left (3 \,{\mathrm e}^{4 x}-2 \,{\mathrm e}^{3 x}-1\right ) {\mathrm e}^{-2 x}}{3} & -{\mathrm e}^{-2 x}+{\mathrm e}^{2 x} & -\frac {\left (-3 \,{\mathrm e}^{4 x}+{\mathrm e}^{3 x}-1\right ) {\mathrm e}^{-2 x}}{3} \end {array}\right ] \\ \square & {} & \textrm {Find a particular solution of the system of ODEs using variation of parameters}\hspace {3pt} \\ {} & \circ & \textrm {Let the particular solution be the fundamental matrix multiplied by}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (x \right )\hspace {3pt}\textrm {and solve for}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (x \right ) \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}\left (x \right )=\Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right ) \\ {} & \circ & \textrm {Take the derivative of the particular solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}^{\prime }\left (x \right )=\Phi ^{\prime }\left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right )+\Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (x \right ) \\ {} & \circ & \textrm {Substitute particular solution and its derivative into the system of ODEs}\hspace {3pt} \\ {} & {} & \Phi ^{\prime }\left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right )+\Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (x \right )=A \cdot \Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right )+{\moverset {\rightarrow }{f}}\left (x \right ) \\ {} & \circ & \textrm {The fundamental matrix has columns that are solutions to the homogeneous system so its derivative follows that of the homogeneous system}\hspace {3pt} \\ {} & {} & A \cdot \Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right )+\Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (x \right )=A \cdot \Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right )+{\moverset {\rightarrow }{f}}\left (x \right ) \\ {} & \circ & \textrm {Cancel like terms}\hspace {3pt} \\ {} & {} & \Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (x \right )={\moverset {\rightarrow }{f}}\left (x \right ) \\ {} & \circ & \textrm {Multiply by the inverse of the fundamental matrix}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{v}}^{\prime }\left (x \right )=\frac {1}{\Phi \left (x \right )}\cdot {\moverset {\rightarrow }{f}}\left (x \right ) \\ {} & \circ & \textrm {Integrate to solve for}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (x \right ) \\ {} & {} & {\moverset {\rightarrow }{v}}\left (x \right )=\int _{0}^{x}\frac {1}{\Phi \left (s \right )}\cdot {\moverset {\rightarrow }{f}}\left (s \right )d s \\ {} & \circ & \textrm {Plug}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (x \right )\hspace {3pt}\textrm {into the equation for the particular solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}\left (x \right )=\Phi \left (x \right )\cdot \left (\int _{0}^{x}\frac {1}{\Phi \left (s \right )}\cdot {\moverset {\rightarrow }{f}}\left (s \right )d s \right ) \\ {} & \circ & \textrm {Plug in the fundamental matrix and the forcing function and compute}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}\left (x \right )=\left [\begin {array}{c} \left (-4 \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{4 x}+1+\left (\left (x +1\right ) \cos \left (x \right )+\left (2-x \right ) \sin \left (x \right )\right ) {\mathrm e}^{x}\right ) {\mathrm e}^{-2 x} \\ -2 \left (2 \,{\mathrm e}^{3 x}-2 \,{\mathrm e}^{4 x}+1+\left (\left (x -1\right ) \cos \left (x \right )+2 \sin \left (x \right )\right ) {\mathrm e}^{x}\right ) {\mathrm e}^{-2 x} \\ 2 \left (-2 \,{\mathrm e}^{3 x}+4 \,{\mathrm e}^{4 x}+2+\left (\left (x -4\right ) \cos \left (x \right )+\left (x +1\right ) \sin \left (x \right )\right ) {\mathrm e}^{x}\right ) {\mathrm e}^{-2 x} \end {array}\right ] \\ \bullet & {} & \textrm {Plug particular solution back into general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}\left (x \right )=c_{1} {\moverset {\rightarrow }{y}}_{1}+c_{2} {\moverset {\rightarrow }{y}}_{2}+c_{3} {\moverset {\rightarrow }{y}}_{3}+\left [\begin {array}{c} \left (-4 \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{4 x}+1+\left (\left (x +1\right ) \cos \left (x \right )+\left (2-x \right ) \sin \left (x \right )\right ) {\mathrm e}^{x}\right ) {\mathrm e}^{-2 x} \\ -2 \left (2 \,{\mathrm e}^{3 x}-2 \,{\mathrm e}^{4 x}+1+\left (\left (x -1\right ) \cos \left (x \right )+2 \sin \left (x \right )\right ) {\mathrm e}^{x}\right ) {\mathrm e}^{-2 x} \\ 2 \left (-2 \,{\mathrm e}^{3 x}+4 \,{\mathrm e}^{4 x}+2+\left (\left (x -4\right ) \cos \left (x \right )+\left (x +1\right ) \sin \left (x \right )\right ) {\mathrm e}^{x}\right ) {\mathrm e}^{-2 x} \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y={\mathrm e}^{-2 x} \left (\left (c_{2} -4\right ) {\mathrm e}^{3 x}+\left (\frac {c_{3}}{4}+2\right ) {\mathrm e}^{4 x}+\left (\left (x +1\right ) \cos \left (x \right )-\left (-2+x \right ) \sin \left (x \right )\right ) {\mathrm e}^{x}+\frac {c_{1}}{4}+1\right ) \end {array} \]
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: 0.0 (sec). Leaf size: 40
dsolve(0*diff(y(x),x$4)+1*diff(y(x),x$3)-1*diff(y(x),x$2)-4*diff(y(x),x)+4*y(x)=exp(-x)*((16+10*x)*cos(x)+(30-10*x)*sin(x)),y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{1} {\mathrm e}^{3 x}+c_{3} {\mathrm e}^{4 x}+\left (\left (x +1\right ) \cos \left (x \right )-\sin \left (x \right ) \left (-2+x \right )\right ) {\mathrm e}^{x}+c_{2} \right ) {\mathrm e}^{-2 x} \]
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 49
DSolve[0*y''''[x]+1*y'''[x]-1*y''[x]-4*y'[x]+4*y[x]==Exp[-x]*((16+10*x)*Cos[x]+(30-10*x)*Sin[x]),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-2 x} \left (-e^x (x-2) \sin (x)+e^x (x+1) \cos (x)+c_2 e^{3 x}+c_3 e^{4 x}+c_1\right ) \]