Internal problem ID [13752]
Internal file name [OUTPUT/12924_Sunday_February_18_2024_08_01_39_AM_32851158/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: 22.12 (c).
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, _missing_y]]
\[ \boxed {y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }=32 \,{\mathrm e}^{4 x}} \] 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 \[ y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime } = 0 \] The characteristic equation is \[ \lambda ^{4}-4 \lambda ^{3} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 4\\ \lambda _2 &= 0\\ \lambda _3 &= 0\\ \lambda _4 &= 0 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=x^{2} c_{3} +c_{2} x +c_{1} +{\mathrm e}^{4 x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= x \\ y_3 &= x^{2} \\ y_4 &= {\mathrm e}^{4 x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime } = 32 \,{\mathrm e}^{4 x} \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ 32 \,{\mathrm e}^{4 x} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{{\mathrm e}^{4 x}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{1, x, x^{2}, {\mathrm e}^{4 x}\} \] Since \({\mathrm e}^{4 x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x \,{\mathrm e}^{4 x}\}] \] 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. \[ y_p = A_{1} x \,{\mathrm e}^{4 x} \] The unknowns \(\{A_{1}\}\) 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 \[ 64 A_{1} {\mathrm e}^{4 x} = 32 \,{\mathrm e}^{4 x} \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = {\frac {1}{2}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {x \,{\mathrm e}^{4 x}}{2} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (x^{2} c_{3} +c_{2} x +c_{1} +{\mathrm e}^{4 x} c_{4}\right ) + \left (\frac {x \,{\mathrm e}^{4 x}}{2}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= x^{2} c_{3} +c_{2} x +c_{1} +{\mathrm e}^{4 x} c_{4} +\frac {x \,{\mathrm e}^{4 x}}{2} \\ \end{align*}
Verification of solutions
\[ y = x^{2} c_{3} +c_{2} x +c_{1} +{\mathrm e}^{4 x} c_{4} +\frac {x \,{\mathrm e}^{4 x}}{2} \] 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) = 4*_b(_a)+32*exp(4*_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`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 27
dsolve(diff(y(x),x$4)-4*diff(y(x),x$3)=32*exp(4*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (32 x +c_{1} -24\right ) {\mathrm e}^{4 x}}{64}+\frac {c_{2} x^{2}}{2}+c_{3} x +c_{4} \]
✓ Solution by Mathematica
Time used: 0.137 (sec). Leaf size: 33
DSolve[y''''[x]-4*y'''[x]==32*Exp[4*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{64} e^{4 x} (32 x-24+c_1)+x (c_4 x+c_3)+c_2 \]