problem Problem 30

Internal problem ID [2750]
Internal ﬁle name [OUTPUT/2242_Sunday_June_05_2022_02_56_03_AM_97144766/index.tex]

Book: Diﬀerential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section: Chapter 8, Linear diﬀerential equations of order n. Section 8.3, The Method of Undetermined Coeﬃcients. page 525
Problem number: Problem 30.
ODE order: 3.
ODE degree: 1.

The type(s) of ODE detected by this program : "higher_order_linear_constant_coeﬃcients_ODE"

\[ \boxed {y^{\prime \prime \prime }+2 y^{\prime \prime }-5 y^{\prime }-6 y=4 x^{2}} \] 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 \[ y^{\prime \prime \prime }+2 y^{\prime \prime }-5 y^{\prime }-6 y = 0 \] The characteristic equation is \[ \lambda ^{3}+2 \lambda ^{2}-5 \lambda -6 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 2\\ \lambda _2 &= -3\\ \lambda _3 &= -1 \end {align*}

Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{-x} c_{1} +c_{2} {\mathrm e}^{-3 x}+c_{3} {\mathrm e}^{2 x} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{-x} \\ y_2 &= {\mathrm e}^{-3 x} \\ y_3 &= {\mathrm e}^{2 x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }+2 y^{\prime \prime }-5 y^{\prime }-6 y = 4 x^{2} \] The particular solution is found using the method of undetermined coeﬃcients. Looking at the RHS of the ode, which is \[ x^{2} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{1, x, x^{2}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{{\mathrm e}^{-3 x}, {\mathrm e}^{-x}, {\mathrm e}^{2 x}\} \] 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. \[ y_p = A_{3} x^{2}+A_{2} x +A_{1} \] 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 \[ -6 A_{3} x^{2}-6 A_{2} x -10 x A_{3}-6 A_{1}-5 A_{2}+4 A_{3} = 4 x^{2} \] Solving for the unknowns by comparing coeﬃcients results in \[ \left [A_{1} = -{\frac {37}{27}}, A_{2} = {\frac {10}{9}}, A_{3} = -{\frac {2}{3}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = -\frac {2}{3} x^{2}+\frac {10}{9} x -\frac {37}{27} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{-x} c_{1} +c_{2} {\mathrm e}^{-3 x}+c_{3} {\mathrm e}^{2 x}\right ) + \left (-\frac {2}{3} x^{2}+\frac {10}{9} x -\frac {37}{27}\right ) \\ \end{align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-x} c_{1} +c_{2} {\mathrm e}^{-3 x}+c_{3} {\mathrm e}^{2 x}-\frac {2 x^{2}}{3}+\frac {10 x}{9}-\frac {37}{27} \\ \end{align*}

Veriﬁcation of solutions

\[ y = {\mathrm e}^{-x} c_{1} +c_{2} {\mathrm e}^{-3 x}+c_{3} {\mathrm e}^{2 x}-\frac {2 x^{2}}{3}+\frac {10 x}{9}-\frac {37}{27} \] Veriﬁed OK.

7.6.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }+2 \frac {d}{d x}y^{\prime }-5 y^{\prime }-6 y=4 x^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \\ \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 )=\frac {d}{d x}y^{\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 )=4 x^{2}-2 y_{3}\left (x \right )+5 y_{2}\left (x \right )+6 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 )=4 x^{2}-2 y_{3}\left (x \right )+5 y_{2}\left (x \right )+6 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 \\ 6 & 5 & -2 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \right )+\left [\begin {array}{c} 0 \\ 0 \\ 4 x^{2} \end {array}\right ] \\ \bullet & {} & \textrm {Define the forcing function}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{f}}\left (x \right )=\left [\begin {array}{c} 0 \\ 0 \\ 4 x^{2} \end {array}\right ] \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 6 & 5 & -2 \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}{9} \\ -\frac {1}{3} \\ 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 [-3, \left [\begin {array}{c} \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}{9} \\ -\frac {1}{3} \\ 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}^{-3 x}}{9} & {\mathrm e}^{-x} & \frac {{\mathrm e}^{2 x}}{4} \\ -\frac {{\mathrm e}^{-3 x}}{3} & -{\mathrm e}^{-x} & \frac {{\mathrm e}^{2 x}}{2} \\ {\mathrm e}^{-3 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 \phi \left (0\right )^{-1} \\ {} & \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}^{-3 x}}{9} & {\mathrm e}^{-x} & \frac {{\mathrm e}^{2 x}}{4} \\ -\frac {{\mathrm e}^{-3 x}}{3} & -{\mathrm e}^{-x} & \frac {{\mathrm e}^{2 x}}{2} \\ {\mathrm e}^{-3 x} & {\mathrm e}^{-x} & {\mathrm e}^{2 x} \end {array}\right ]\cdot \left [\begin {array}{ccc} \frac {1}{9} & 1 & \frac {1}{4} \\ -\frac {1}{3} & -1 & \frac {1}{2} \\ 1 & 1 & 1 \end {array}\right ]^{-1} \\ {} & \circ & \textrm {Evaluate and simplify to get the fundamental matrix}\hspace {3pt} \\ {} & {} & \Phi \left (x \right )=\left [\begin {array}{ccc} \frac {\left ({\mathrm e}^{5 x}+5 \,{\mathrm e}^{2 x}-1\right ) {\mathrm e}^{-3 x}}{5} & -\frac {\left (-8 \,{\mathrm e}^{5 x}+5 \,{\mathrm e}^{2 x}+3\right ) {\mathrm e}^{-3 x}}{30} & -\frac {\left (-2 \,{\mathrm e}^{5 x}+5 \,{\mathrm e}^{2 x}-3\right ) {\mathrm e}^{-3 x}}{30} \\ -\frac {\left (-2 \,{\mathrm e}^{5 x}+5 \,{\mathrm e}^{2 x}-3\right ) {\mathrm e}^{-3 x}}{5} & \frac {\left (16 \,{\mathrm e}^{5 x}+5 \,{\mathrm e}^{2 x}+9\right ) {\mathrm e}^{-3 x}}{30} & \frac {\left (4 \,{\mathrm e}^{5 x}+5 \,{\mathrm e}^{2 x}-9\right ) {\mathrm e}^{-3 x}}{30} \\ \frac {\left (4 \,{\mathrm e}^{5 x}+5 \,{\mathrm e}^{2 x}-9\right ) {\mathrm e}^{-3 x}}{5} & -\frac {\left (-32 \,{\mathrm e}^{5 x}+5 \,{\mathrm e}^{2 x}+27\right ) {\mathrm e}^{-3 x}}{30} & -\frac {\left (-8 \,{\mathrm e}^{5 x}+5 \,{\mathrm e}^{2 x}-27\right ) {\mathrm e}^{-3 x}}{30} \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 )=\Phi \left (x \right )^{-1}\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}\Phi \left (s \right )^{-1}\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}\Phi \left (s \right )^{-1}\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} \frac {{\mathrm e}^{-3 x} \left (\left (-18 x^{2}+30 x -37\right ) {\mathrm e}^{3 x}+36 \,{\mathrm e}^{2 x}+\frac {9 \,{\mathrm e}^{5 x}}{5}-\frac {4}{5}\right )}{27} \\ -\frac {2 \left (-3 \,{\mathrm e}^{5 x}+30 x \,{\mathrm e}^{3 x}-25 \,{\mathrm e}^{3 x}+30 \,{\mathrm e}^{2 x}-2\right ) {\mathrm e}^{-3 x}}{45} \\ \frac {4 \left ({\mathrm e}^{5 x}-5 \,{\mathrm e}^{3 x}+5 \,{\mathrm e}^{2 x}-1\right ) {\mathrm e}^{-3 x}}{15} \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} \frac {{\mathrm e}^{-3 x} \left (\left (-18 x^{2}+30 x -37\right ) {\mathrm e}^{3 x}+36 \,{\mathrm e}^{2 x}+\frac {9 \,{\mathrm e}^{5 x}}{5}-\frac {4}{5}\right )}{27} \\ -\frac {2 \left (-3 \,{\mathrm e}^{5 x}+30 x \,{\mathrm e}^{3 x}-25 \,{\mathrm e}^{3 x}+30 \,{\mathrm e}^{2 x}-2\right ) {\mathrm e}^{-3 x}}{45} \\ \frac {4 \left ({\mathrm e}^{5 x}-5 \,{\mathrm e}^{3 x}+5 \,{\mathrm e}^{2 x}-1\right ) {\mathrm e}^{-3 x}}{15} \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {{\mathrm e}^{-3 x} \left (\left (-18 x^{2}+30 x -37\right ) {\mathrm e}^{3 x}+27 c_{2} {\mathrm e}^{2 x}+\frac {27 c_{3} {\mathrm e}^{5 x}}{4}+3 c_{1} +36 \,{\mathrm e}^{2 x}+\frac {9 \,{\mathrm e}^{5 x}}{5}-\frac {4}{5}\right )}{27} \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`

\[ y \left (x \right ) = \frac {\left (-18 x^{2}+30 x -37\right ) {\mathrm e}^{-3 x} {\mathrm e}^{3 x}}{27}+\left (c_{2} {\mathrm e}^{2 x}+c_{3} {\mathrm e}^{5 x}+c_{1} \right ) {\mathrm e}^{-3 x} \]

\[ y(x)\to -\frac {2 x^2}{3}+\frac {10 x}{9}+c_1 e^{-3 x}+c_2 e^{-x}+c_3 e^{2 x}-\frac {37}{27} \]

7.6 problem Problem 30

7.6.1 Maple step by step solution