Internal problem ID [5914]
Internal file name [OUTPUT/5162_Sunday_June_05_2022_03_26_34_PM_30795132/index.tex
]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 1.3 Introduction– Linear equations of First Order. Page 38
Problem number: 1 (d).
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, _quadrature]]
\[ \boxed {y^{\prime \prime \prime }=x^{2}} \] 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 \[ y^{\prime \prime \prime } = 0 \] The characteristic equation is \[ \lambda ^{3} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 0\\ \lambda _3 &= 0 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{3} x^{2}+c_{2} x +c_{1} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= x \\ y_3 &= x^{2} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime } = x^{2} \] 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 \[ \{1, x, x^{2}\} \] 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 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. \[ y_p = A_{3} x^{5}+A_{2} x^{4}+A_{1} x^{3} \] 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 \[ 60 x^{2} A_{3}+24 x A_{2}+6 A_{1} = x^{2} \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = 0, A_{2} = 0, A_{3} = {\frac {1}{60}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {x^{5}}{60} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{3} x^{2}+c_{2} x +c_{1}\right ) + \left (\frac {x^{5}}{60}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{3} x^{2}+c_{2} x +c_{1} +\frac {1}{60} x^{5} \\ \end{align*}
Verification of solutions
\[ y = c_{3} x^{2}+c_{2} x +c_{1} +\frac {1}{60} x^{5} \] Verified OK.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 20
dsolve(diff(y(x),x$3)=x^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {1}{60} x^{5}+\frac {1}{2} c_{1} x^{2}+c_{2} x +c_{3} \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 25
DSolve[y'''[x]==x^2,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^5}{60}+c_3 x^2+c_2 x+c_1 \]