problem 5

Internal problem ID [2238]
Internal ﬁle name [OUTPUT/2238_Monday_February_26_2024_09_18_37_AM_94999196/index.tex]

Book: Diﬀerential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section: Exercise 24, page 109
Problem number: 5.
ODE order: 5.
ODE degree: 1.

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

\[ \boxed {y^{\left (5\right )}+y^{\prime \prime \prime \prime }=x^{2}} \] 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 \[ y^{\left (5\right )}+y^{\prime \prime \prime \prime } = 0 \] The characteristic equation is \[ \lambda ^{5}+\lambda ^{4} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -1\\ \lambda _2 &= 0\\ \lambda _3 &= 0\\ \lambda _4 &= 0\\ \lambda _5 &= 0 \end {align*}

Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{-x}+c_{2} +c_{3} x +c_{4} x^{2}+c_{5} x^{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{-x} \\ y_2 &= 1 \\ y_3 &= x \\ y_4 &= x^{2} \\ y_5 &= x^{3} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\left (5\right )}+y^{\prime \prime \prime \prime } = x^{2} \] 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 \[ \{1, x, x^{2}, x^{3}, {\mathrm e}^{-x}\} \] Since \(1\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x, x^{2}, x^{3}\}] \] Since \(x\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x^{2}, x^{3}, x^{4}\}] \] Since \(x^{2}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x^{3}, x^{4}, x^{5}\}] \] Since \(x^{3}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x^{4}, x^{5}, x^{6}\}] \] Since there was duplication between the basis functions in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis function in the above updated UC_set. \[ y_p = A_{3} x^{6}+A_{2} x^{5}+A_{1} x^{4} \] 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 \[ 360 x^{2} A_{3}+120 x A_{2}+720 x A_{3}+24 A_{1}+120 A_{2} = x^{2} \] Solving for the unknowns by comparing coeﬃcients results in \[ \left [A_{1} = {\frac {1}{12}}, A_{2} = -{\frac {1}{60}}, A_{3} = {\frac {1}{360}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {1}{360} x^{6}-\frac {1}{60} x^{5}+\frac {1}{12} x^{4} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} {\mathrm e}^{-x}+c_{2} +c_{3} x +c_{4} x^{2}+c_{5} x^{3}\right ) + \left (\frac {1}{360} x^{6}-\frac {1}{60} x^{5}+\frac {1}{12} x^{4}\right ) \\ \end{align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{-x}+c_{2} +c_{3} x +c_{4} x^{2}+c_{5} x^{3}+\frac {x^{6}}{360}-\frac {x^{5}}{60}+\frac {x^{4}}{12} \\ \end{align*}

Veriﬁcation of solutions

\[ y = c_{1} {\mathrm e}^{-x}+c_{2} +c_{3} x +c_{4} x^{2}+c_{5} x^{3}+\frac {x^{6}}{360}-\frac {x^{5}}{60}+\frac {x^{4}}{12} \] Veriﬁed OK.

Maple trace

`Methods for high order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
-> Calling odsolve with the ODE`, diff(_b(_a), _a) = _a^2-_b(_a), _b(_a)`   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   --- Trying classification methods --- 
   trying a quadrature 
   trying 1st order linear 
   <- 1st order linear successful 
<- high order exact linear fully integrable successful`

\[ y \left (x \right ) = \frac {x^{6}}{360}+\frac {x^{4}}{12}-\frac {x^{5}}{60}+\frac {c_{2} x^{3}}{6}+\frac {c_{3} x^{2}}{2}+{\mathrm e}^{-x} c_{1} +c_{4} x +c_{5} \]

\[ y(x)\to \frac {x^6}{360}-\frac {x^5}{60}+\frac {x^4}{12}+c_5 x^3+c_4 x^2+c_3 x+c_1 e^{-x}+c_2 \]

15.5 problem 5