problem 22.13 (a)

Internal problem ID [13757]
Internal ﬁle name [OUTPUT/12929_Sunday_February_18_2024_08_01_40_AM_76081755/index.tex]

Book: Ordinary Diﬀerential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section: Chapter 22. Method of undetermined coeﬃcients. Additional exercises page 412
Problem number: 22.13 (a).
ODE order: 5.
ODE degree: 1.

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

\[ \boxed {y^{\left (5\right )}+18 y^{\prime \prime \prime }+81 y^{\prime }=x^{2} {\mathrm e}^{3 x}} \] 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 \[ y^{\left (5\right )}+18 y^{\prime \prime \prime }+81 y^{\prime } = 0 \] The characteristic equation is \[ \lambda ^{5}+18 \lambda ^{3}+81 \lambda = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 3 i\\ \lambda _3 &= -3 i\\ \lambda _4 &= 3 i\\ \lambda _5 &= -3 i \end {align*}

Therefore the homogeneous solution is \[ y_h(x)=c_{1} +{\mathrm e}^{-3 i x} c_{2} +x \,{\mathrm e}^{-3 i x} c_{3} +{\mathrm e}^{3 i x} c_{4} +x \,{\mathrm e}^{3 i x} c_{5} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= {\mathrm e}^{-3 i x} \\ y_3 &= x \,{\mathrm e}^{-3 i x} \\ y_4 &= {\mathrm e}^{3 i x} \\ y_5 &= x \,{\mathrm e}^{3 i x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\left (5\right )}+18 y^{\prime \prime \prime }+81 y^{\prime } = x^{2} {\mathrm e}^{3 x} \] The particular solution is found using the method of undetermined coeﬃcients. Looking at the RHS of the ode, which is \[ x^{2} {\mathrm e}^{3 x} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{x^{2} {\mathrm e}^{3 x}, {\mathrm e}^{3 x} x, {\mathrm e}^{3 x}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{1, x \,{\mathrm e}^{-3 i x}, x \,{\mathrm e}^{3 i x}, {\mathrm e}^{-3 i x}, {\mathrm e}^{3 i x}\} \] 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. \[ y_p = A_{1} x^{2} {\mathrm e}^{3 x}+A_{2} {\mathrm e}^{3 x} x +A_{3} {\mathrm e}^{3 x} \] 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 \[ 864 A_{1} {\mathrm e}^{3 x}+1944 A_{1} x \,{\mathrm e}^{3 x}+972 A_{1} x^{2} {\mathrm e}^{3 x}+972 A_{2} {\mathrm e}^{3 x} x +972 A_{2} {\mathrm e}^{3 x}+972 A_{3} {\mathrm e}^{3 x} = x^{2} {\mathrm e}^{3 x} \] Solving for the unknowns by comparing coeﬃcients results in \[ \left [A_{1} = {\frac {1}{972}}, A_{2} = -{\frac {1}{486}}, A_{3} = {\frac {5}{4374}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {x^{2} {\mathrm e}^{3 x}}{972}-\frac {{\mathrm e}^{3 x} x}{486}+\frac {5 \,{\mathrm e}^{3 x}}{4374} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} +{\mathrm e}^{-3 i x} c_{2} +x \,{\mathrm e}^{-3 i x} c_{3} +{\mathrm e}^{3 i x} c_{4} +x \,{\mathrm e}^{3 i x} c_{5}\right ) + \left (\frac {x^{2} {\mathrm e}^{3 x}}{972}-\frac {{\mathrm e}^{3 x} x}{486}+\frac {5 \,{\mathrm e}^{3 x}}{4374}\right ) \\ \end{align*} Which simpliﬁes to \[ y = \left (c_{3} x +c_{2} \right ) {\mathrm e}^{-3 i x}+\left (c_{5} x +c_{4} \right ) {\mathrm e}^{3 i x}+c_{1} +\frac {x^{2} {\mathrm e}^{3 x}}{972}-\frac {{\mathrm e}^{3 x} x}{486}+\frac {5 \,{\mathrm e}^{3 x}}{4374} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \left (c_{3} x +c_{2} \right ) {\mathrm e}^{-3 i x}+\left (c_{5} x +c_{4} \right ) {\mathrm e}^{3 i x}+c_{1} +\frac {x^{2} {\mathrm e}^{3 x}}{972}-\frac {{\mathrm e}^{3 x} x}{486}+\frac {5 \,{\mathrm e}^{3 x}}{4374} \\ \end{align*}

Veriﬁcation of solutions

\[ y = \left (c_{3} x +c_{2} \right ) {\mathrm e}^{-3 i x}+\left (c_{5} x +c_{4} \right ) {\mathrm e}^{3 i x}+c_{1} +\frac {x^{2} {\mathrm e}^{3 x}}{972}-\frac {{\mathrm e}^{3 x} x}{486}+\frac {5 \,{\mathrm e}^{3 x}}{4374} \] 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: 5; linear nonhomogeneous with symmetry [0,1] 
-> Calling odsolve with the ODE`, diff(diff(diff(diff(_b(_a), _a), _a), _a), _a) = _a^2*exp(3*_a)-18*(diff(diff(_b(_a), _a), _a))-81 
   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 
<- differential order: 5; linear nonhomogeneous with symmetry [0,1] successful`

\[ y \left (x \right ) = \frac {\left (-3 c_{4} x -3 c_{2} +c_{3} \right ) \cos \left (3 x \right )}{9}+\frac {\left (9 x^{2}-18 x +10\right ) {\mathrm e}^{3 x}}{8748}+\frac {\left (3 c_{3} x +3 c_{1} +c_{4} \right ) \sin \left (3 x \right )}{9}+c_{5} \]

\[ y(x)\to \frac {e^{3 x} \left (9 x^2-18 x+10\right )}{8748}+\frac {1}{9} (c_2-3 (c_4 x+c_3)) \cos (3 x)+\frac {1}{9} (3 c_2 x+3 c_1+c_4) \sin (3 x)+c_5 \]

15.63 problem 22.13 (a)