Internal problem ID [1510]
Internal file name [OUTPUT/1511_Sunday_June_05_2022_02_20_08_AM_9104730/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 13.
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, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime \prime \prime }+3 y^{\prime \prime \prime }-3 y^{\prime \prime }-7 y^{\prime }+6 y=-3 \,{\mathrm e}^{-x} \left (-8 x^{2}+8 x +12\right )} \] 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 \[ y^{\prime \prime \prime \prime }+3 y^{\prime \prime \prime }-3 y^{\prime \prime }-7 y^{\prime }+6 y = 0 \] The characteristic equation is \[ \lambda ^{4}+3 \lambda ^{3}-3 \lambda ^{2}-7 \lambda +6 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -3\\ \lambda _2 &= -2\\ \lambda _3 &= 1\\ \lambda _4 &= 1 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{x}+x \,{\mathrm e}^{x} c_{3} +{\mathrm e}^{-3 x} c_{4} \] 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 &= x \,{\mathrm e}^{x} \\ y_4 &= {\mathrm e}^{-3 x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }+3 y^{\prime \prime \prime }-3 y^{\prime \prime }-7 y^{\prime }+6 y = -3 \,{\mathrm e}^{-x} \left (-8 x^{2}+8 x +12\right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ -3 \,{\mathrm e}^{-x} \left (-8 x^{2}+8 x +12\right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{x \,{\mathrm e}^{-x}, x^{2} {\mathrm e}^{-x}, {\mathrm e}^{-x}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{x \,{\mathrm e}^{x}, {\mathrm e}^{x}, {\mathrm e}^{-3 x}, {\mathrm e}^{-2 x}\} \] 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. \[ y_p = A_{1} x \,{\mathrm e}^{-x}+A_{2} x^{2} {\mathrm e}^{-x}+A_{3} {\mathrm e}^{-x} \] The unknowns \(\{A_{1}, A_{2}, A_{3}\}\) 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 \[ 8 A_{3} {\mathrm e}^{-x}+8 A_{2} x \,{\mathrm e}^{-x}+4 A_{1} {\mathrm e}^{-x}-12 A_{2} {\mathrm e}^{-x}+8 A_{1} x \,{\mathrm e}^{-x}+8 A_{2} x^{2} {\mathrm e}^{-x} = -3 \,{\mathrm e}^{-x} \left (-8 x^{2}+8 x +12\right ) \] Solving for the unknowns by comparing coefficients results in \[ [A_{1} = -6, A_{2} = 3, A_{3} = 3] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = -6 x \,{\mathrm e}^{-x}+3 x^{2} {\mathrm e}^{-x}+3 \,{\mathrm e}^{-x} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{x}+x \,{\mathrm e}^{x} c_{3} +{\mathrm e}^{-3 x} c_{4}\right ) + \left (-6 x \,{\mathrm e}^{-x}+3 x^{2} {\mathrm e}^{-x}+3 \,{\mathrm e}^{-x}\right ) \\ \end{align*} Which simplifies to \[ y = {\mathrm e}^{-3 x} \left (\left (c_{3} x +c_{2} \right ) {\mathrm e}^{4 x}+c_{1} {\mathrm e}^{x}+c_{4} \right )-6 x \,{\mathrm e}^{-x}+3 x^{2} {\mathrm e}^{-x}+3 \,{\mathrm e}^{-x} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-3 x} \left (\left (c_{3} x +c_{2} \right ) {\mathrm e}^{4 x}+c_{1} {\mathrm e}^{x}+c_{4} \right )-6 x \,{\mathrm e}^{-x}+3 x^{2} {\mathrm e}^{-x}+3 \,{\mathrm e}^{-x} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{-3 x} \left (\left (c_{3} x +c_{2} \right ) {\mathrm e}^{4 x}+c_{1} {\mathrm e}^{x}+c_{4} \right )-6 x \,{\mathrm e}^{-x}+3 x^{2} {\mathrm e}^{-x}+3 \,{\mathrm e}^{-x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime \prime }+3 y^{\prime \prime \prime }-3 y^{\prime \prime }-7 y^{\prime }+6 y=-3 \,{\mathrm e}^{-x} \left (-8 x^{2}+8 x +12\right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & y^{\prime \prime \prime \prime } \\ \bullet & {} & \textrm {Isolate 4th derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime \prime }=-6 y+24 x^{2} {\mathrm e}^{-x}-24 x \,{\mathrm e}^{-x}-3 y^{\prime \prime \prime }+3 y^{\prime \prime }+7 y^{\prime }-36 \,{\mathrm e}^{-x} \\ \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 \prime }+3 y^{\prime \prime \prime }-3 y^{\prime \prime }-7 y^{\prime }+6 y=12 \,{\mathrm e}^{-x} \left (2 x^{2}-2 x -3\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 {Define new variable}\hspace {3pt} y_{4}\left (x \right ) \\ {} & {} & y_{4}\left (x \right )=y^{\prime \prime \prime } \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} y_{4}^{\prime }\left (x \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{4}^{\prime }\left (x \right )=24 x^{2} {\mathrm e}^{-x}-24 x \,{\mathrm e}^{-x}-3 y_{4}\left (x \right )+3 y_{3}\left (x \right )+7 y_{2}\left (x \right )-6 y_{1}\left (x \right )-36 \,{\mathrm e}^{-x} \\ & {} & \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_{4}\left (x \right )=y_{3}^{\prime }\left (x \right ), y_{4}^{\prime }\left (x \right )=24 x^{2} {\mathrm e}^{-x}-24 x \,{\mathrm e}^{-x}-3 y_{4}\left (x \right )+3 y_{3}\left (x \right )+7 y_{2}\left (x \right )-6 y_{1}\left (x \right )-36 \,{\mathrm e}^{-x}\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 ) \\ y_{4}\left (x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -6 & 7 & 3 & -3 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \right )+\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 24 x^{2} {\mathrm e}^{-x}-24 x \,{\mathrm e}^{-x}-36 \,{\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 \\ 0 \\ 24 x^{2} {\mathrm e}^{-x}-24 x \,{\mathrm e}^{-x}-36 \,{\mathrm e}^{-x} \end {array}\right ] \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -6 & 7 & 3 & -3 \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 [-3, \left [\begin {array}{c} -\frac {1}{27} \\ \frac {1}{9} \\ -\frac {1}{3} \\ 1 \end {array}\right ]\right ], \left [-2, \left [\begin {array}{c} -\frac {1}{8} \\ \frac {1}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ]\right ], \left [1, \left [\begin {array}{c} 1 \\ 1 \\ 1 \\ 1 \end {array}\right ]\right ], \left [1, \left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-3, \left [\begin {array}{c} -\frac {1}{27} \\ \frac {1}{9} \\ -\frac {1}{3} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{-3 x}\cdot \left [\begin {array}{c} -\frac {1}{27} \\ \frac {1}{9} \\ -\frac {1}{3} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-2, \left [\begin {array}{c} -\frac {1}{8} \\ \frac {1}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{2}={\mathrm e}^{-2 x}\cdot \left [\begin {array}{c} -\frac {1}{8} \\ \frac {1}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair, with eigenvalue of algebraic multiplicity 2}\hspace {3pt} \\ {} & {} & \left [1, \left [\begin {array}{c} 1 \\ 1 \\ 1 \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {First solution from eigenvalue}\hspace {3pt} 1 \\ {} & {} & {\moverset {\rightarrow }{y}}_{3}\left (x \right )={\mathrm e}^{x}\cdot \left [\begin {array}{c} 1 \\ 1 \\ 1 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Form of the 2nd homogeneous solution where}\hspace {3pt} {\moverset {\rightarrow }{p}}\hspace {3pt}\textrm {is to be solved for,}\hspace {3pt} \lambda =1\hspace {3pt}\textrm {is the eigenvalue, and}\hspace {3pt} {\moverset {\rightarrow }{v}}\hspace {3pt}\textrm {is the eigenvector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{4}\left (x \right )={\mathrm e}^{\lambda x} \left (x {\moverset {\rightarrow }{v}}+{\moverset {\rightarrow }{p}}\right ) \\ \bullet & {} & \textrm {Note that the}\hspace {3pt} x \hspace {3pt}\textrm {multiplying}\hspace {3pt} {\moverset {\rightarrow }{v}}\hspace {3pt}\textrm {makes this solution linearly independent to the 1st solution obtained from}\hspace {3pt} \lambda =1 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} {\moverset {\rightarrow }{y}}_{4}\left (x \right )\hspace {3pt}\textrm {into the homogeneous system}\hspace {3pt} \\ {} & {} & \lambda \,{\mathrm e}^{\lambda x} \left (x {\moverset {\rightarrow }{v}}+{\moverset {\rightarrow }{p}}\right )+{\mathrm e}^{\lambda x} {\moverset {\rightarrow }{v}}=\left ({\mathrm e}^{\lambda x} A \right )\cdot \left (x {\moverset {\rightarrow }{v}}+{\moverset {\rightarrow }{p}}\right ) \\ \bullet & {} & \textrm {Use the fact that}\hspace {3pt} {\moverset {\rightarrow }{v}}\hspace {3pt}\textrm {is an eigenvector of}\hspace {3pt} A \\ {} & {} & \lambda \,{\mathrm e}^{\lambda x} \left (x {\moverset {\rightarrow }{v}}+{\moverset {\rightarrow }{p}}\right )+{\mathrm e}^{\lambda x} {\moverset {\rightarrow }{v}}={\mathrm e}^{\lambda x} \left (\lambda x {\moverset {\rightarrow }{v}}+A \cdot {\moverset {\rightarrow }{p}}\right ) \\ \bullet & {} & \textrm {Simplify equation}\hspace {3pt} \\ {} & {} & \lambda {\moverset {\rightarrow }{p}}+{\moverset {\rightarrow }{v}}=A \cdot {\moverset {\rightarrow }{p}} \\ \bullet & {} & \textrm {Make use of the identity matrix}\hspace {3pt} \mathrm {I} \\ {} & {} & \left (\lambda \cdot I \right )\cdot {\moverset {\rightarrow }{p}}+{\moverset {\rightarrow }{v}}=A \cdot {\moverset {\rightarrow }{p}} \\ \bullet & {} & \textrm {Condition}\hspace {3pt} {\moverset {\rightarrow }{p}}\hspace {3pt}\textrm {must meet for}\hspace {3pt} {\moverset {\rightarrow }{y}}_{4}\left (x \right )\hspace {3pt}\textrm {to be a solution to the homogeneous system}\hspace {3pt} \\ {} & {} & \left (A -\lambda \cdot I \right )\cdot {\moverset {\rightarrow }{p}}={\moverset {\rightarrow }{v}} \\ \bullet & {} & \textrm {Choose}\hspace {3pt} {\moverset {\rightarrow }{p}}\hspace {3pt}\textrm {to use in the second solution to the homogeneous system from eigenvalue}\hspace {3pt} 1 \\ {} & {} & \left (\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -6 & 7 & 3 & -3 \end {array}\right ]-1\cdot \left [\begin {array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end {array}\right ]\right )\cdot {\moverset {\rightarrow }{p}}=\left [\begin {array}{c} 1 \\ 1 \\ 1 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Choice of}\hspace {3pt} {\moverset {\rightarrow }{p}} \\ {} & {} & {\moverset {\rightarrow }{p}}=\left [\begin {array}{c} -1 \\ 0 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {Second solution from eigenvalue}\hspace {3pt} 1 \\ {} & {} & {\moverset {\rightarrow }{y}}_{4}\left (x \right )={\mathrm e}^{x}\cdot \left (x \cdot \left [\begin {array}{c} 1 \\ 1 \\ 1 \\ 1 \end {array}\right ]+\left [\begin {array}{c} -1 \\ 0 \\ 0 \\ 0 \end {array}\right ]\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}\left (x \right )+c_{4} {\moverset {\rightarrow }{y}}_{4}\left (x \right )+{\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}{cccc} -\frac {{\mathrm e}^{-3 x}}{27} & -\frac {{\mathrm e}^{-2 x}}{8} & {\mathrm e}^{x} & \left (x -1\right ) {\mathrm e}^{x} \\ \frac {{\mathrm e}^{-3 x}}{9} & \frac {{\mathrm e}^{-2 x}}{4} & {\mathrm e}^{x} & x \,{\mathrm e}^{x} \\ -\frac {{\mathrm e}^{-3 x}}{3} & -\frac {{\mathrm e}^{-2 x}}{2} & {\mathrm e}^{x} & x \,{\mathrm e}^{x} \\ {\mathrm e}^{-3 x} & {\mathrm e}^{-2 x} & {\mathrm e}^{x} & x \,{\mathrm e}^{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}{cccc} -\frac {{\mathrm e}^{-3 x}}{27} & -\frac {{\mathrm e}^{-2 x}}{8} & {\mathrm e}^{x} & \left (x -1\right ) {\mathrm e}^{x} \\ \frac {{\mathrm e}^{-3 x}}{9} & \frac {{\mathrm e}^{-2 x}}{4} & {\mathrm e}^{x} & x \,{\mathrm e}^{x} \\ -\frac {{\mathrm e}^{-3 x}}{3} & -\frac {{\mathrm e}^{-2 x}}{2} & {\mathrm e}^{x} & x \,{\mathrm e}^{x} \\ {\mathrm e}^{-3 x} & {\mathrm e}^{-2 x} & {\mathrm e}^{x} & x \,{\mathrm e}^{x} \end {array}\right ]\cdot \frac {1}{\left [\begin {array}{cccc} -\frac {1}{27} & -\frac {1}{8} & 1 & -1 \\ \frac {1}{9} & \frac {1}{4} & 1 & 0 \\ -\frac {1}{3} & -\frac {1}{2} & 1 & 0 \\ 1 & 1 & 1 & 0 \end {array}\right ]} \\ {} & \circ & \textrm {Evaluate and simplify to get the fundamental matrix}\hspace {3pt} \\ {} & {} & \Phi \left (x \right )=\left [\begin {array}{cccc} -\left (x -1\right ) {\mathrm e}^{x} & \frac {\left (\left (2+x \right ) {\mathrm e}^{4 x}-3 \,{\mathrm e}^{x}+1\right ) {\mathrm e}^{-3 x}}{6} & \frac {\left (8 \,{\mathrm e}^{4 x} x -3 \,{\mathrm e}^{4 x}+4 \,{\mathrm e}^{x}-1\right ) {\mathrm e}^{-3 x}}{12} & \frac {\left (2 \,{\mathrm e}^{4 x} x -{\mathrm e}^{4 x}+2 \,{\mathrm e}^{x}-1\right ) {\mathrm e}^{-3 x}}{12} \\ -x \,{\mathrm e}^{x} & \frac {{\mathrm e}^{-3 x} \left (\left (x +3\right ) {\mathrm e}^{4 x}+6 \,{\mathrm e}^{x}-3\right )}{6} & \frac {\left (8 \,{\mathrm e}^{4 x} x +5 \,{\mathrm e}^{4 x}-8 \,{\mathrm e}^{x}+3\right ) {\mathrm e}^{-3 x}}{12} & \frac {\left (2 \,{\mathrm e}^{4 x} x +{\mathrm e}^{4 x}-4 \,{\mathrm e}^{x}+3\right ) {\mathrm e}^{-3 x}}{12} \\ -x \,{\mathrm e}^{x} & \frac {{\mathrm e}^{-3 x} \left (\left (x +3\right ) {\mathrm e}^{4 x}-12 \,{\mathrm e}^{x}+9\right )}{6} & \frac {\left (8 \,{\mathrm e}^{4 x} x +5 \,{\mathrm e}^{4 x}+16 \,{\mathrm e}^{x}-9\right ) {\mathrm e}^{-3 x}}{12} & \frac {\left (2 \,{\mathrm e}^{4 x} x +{\mathrm e}^{4 x}+8 \,{\mathrm e}^{x}-9\right ) {\mathrm e}^{-3 x}}{12} \\ -x \,{\mathrm e}^{x} & \frac {{\mathrm e}^{-3 x} \left (\left (x +3\right ) {\mathrm e}^{4 x}+24 \,{\mathrm e}^{x}-27\right )}{6} & \frac {\left (8 \,{\mathrm e}^{4 x} x +5 \,{\mathrm e}^{4 x}-32 \,{\mathrm e}^{x}+27\right ) {\mathrm e}^{-3 x}}{12} & \frac {\left (2 \,{\mathrm e}^{4 x} x +{\mathrm e}^{4 x}-16 \,{\mathrm e}^{x}+27\right ) {\mathrm e}^{-3 x}}{12} \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} {\mathrm e}^{-3 x} \left (\left (5 x^{2}-9 x +4\right ) {\mathrm e}^{2 x}-3 \,{\mathrm e}^{4 x} x -6 \,{\mathrm e}^{x}+\frac {5 \,{\mathrm e}^{4 x}}{2}-\frac {1}{2}\right ) \\ {\mathrm e}^{-3 x} \left (\left (-5 x^{2}+19 x -13\right ) {\mathrm e}^{2 x}-3 \,{\mathrm e}^{4 x} x +12 \,{\mathrm e}^{x}-\frac {{\mathrm e}^{4 x}}{2}+\frac {3}{2}\right ) \\ {\mathrm e}^{-3 x} \left (\left (7 x^{2}-29 x +29\right ) {\mathrm e}^{2 x}-3 \,{\mathrm e}^{4 x} x -24 \,{\mathrm e}^{x}-\frac {{\mathrm e}^{4 x}}{2}-\frac {9}{2}\right ) \\ {\mathrm e}^{-3 x} \left (\left (-5 x^{2}+43 x -61\right ) {\mathrm e}^{2 x}-3 \,{\mathrm e}^{4 x} x +48 \,{\mathrm e}^{x}-\frac {{\mathrm e}^{4 x}}{2}+\frac {27}{2}\right ) \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 (x \right )+c_{4} {\moverset {\rightarrow }{y}}_{4}\left (x \right )+\left [\begin {array}{c} {\mathrm e}^{-3 x} \left (\left (5 x^{2}-9 x +4\right ) {\mathrm e}^{2 x}-3 \,{\mathrm e}^{4 x} x -6 \,{\mathrm e}^{x}+\frac {5 \,{\mathrm e}^{4 x}}{2}-\frac {1}{2}\right ) \\ {\mathrm e}^{-3 x} \left (\left (-5 x^{2}+19 x -13\right ) {\mathrm e}^{2 x}-3 \,{\mathrm e}^{4 x} x +12 \,{\mathrm e}^{x}-\frac {{\mathrm e}^{4 x}}{2}+\frac {3}{2}\right ) \\ {\mathrm e}^{-3 x} \left (\left (7 x^{2}-29 x +29\right ) {\mathrm e}^{2 x}-3 \,{\mathrm e}^{4 x} x -24 \,{\mathrm e}^{x}-\frac {{\mathrm e}^{4 x}}{2}-\frac {9}{2}\right ) \\ {\mathrm e}^{-3 x} \left (\left (-5 x^{2}+43 x -61\right ) {\mathrm e}^{2 x}-3 \,{\mathrm e}^{4 x} x +48 \,{\mathrm e}^{x}-\frac {{\mathrm e}^{4 x}}{2}+\frac {27}{2}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=5 \,{\mathrm e}^{-3 x} \left (-\frac {1}{10}+\left (\frac {1}{2}+\frac {\left (c_{4} -3\right ) x}{5}+\frac {c_{3}}{5}-\frac {c_{4}}{5}\right ) {\mathrm e}^{4 x}+\left (x^{2}-\frac {9}{5} x +\frac {4}{5}\right ) {\mathrm e}^{2 x}+\frac {\left (-6-\frac {c_{2}}{8}\right ) {\mathrm e}^{x}}{5}-\frac {c_{1}}{135}\right ) \end {array} \]
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] trying high order linear exact nonhomogeneous trying differential order: 4; missing the dependent variable checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 40
dsolve(diff(y(x),x$4)+3*diff(y(x),x$3)-3*diff(y(x),x$2)-7*diff(y(x),x)+6*y(x)=-3*exp(-x)*(12+8*x-8*x^2),y(x), singsol=all)
\[ y \left (x \right ) = 3 \left (\left (x -1\right )^{2} {\mathrm e}^{2 x}+\frac {\left (c_{4} x +c_{1} \right ) {\mathrm e}^{4 x}}{3}+\frac {c_{3} {\mathrm e}^{x}}{3}+\frac {c_{2}}{3}\right ) {\mathrm e}^{-3 x} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 45
DSolve[y''''[x]+3*y'''[x]-3*y''[x]-7*y'[x]+6*y[x]==-3*Exp[-x]*(12+8*x-8*x^2),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-3 x} \left (3 e^{2 x} (x-1)^2+c_2 e^x+e^{4 x} (c_4 x+c_3)+c_1\right ) \]