Internal problem ID [13556]
Internal file name [OUTPUT/12728_Saturday_February_17_2024_08_44_13_AM_28168364/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 14. Higher order equations and the reduction of order method. Additional
exercises page 277
Problem number: 14.5 (b).
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, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime \prime }-9 y^{\prime \prime }+27 y^{\prime }-27 y={\mathrm e}^{3 x} \sin \left (x \right )} \] 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 \[ y^{\prime \prime \prime }-9 y^{\prime \prime }+27 y^{\prime }-27 y = 0 \] The characteristic equation is \[ \lambda ^{3}-9 \lambda ^{2}+27 \lambda -27 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 3\\ \lambda _2 &= 3\\ \lambda _3 &= 3 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{3 x} c_{1} +x \,{\mathrm e}^{3 x} c_{2} +x^{2} {\mathrm e}^{3 x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{3 x} \\ y_2 &= x \,{\mathrm e}^{3 x} \\ y_3 &= x^{2} {\mathrm e}^{3 x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }-9 y^{\prime \prime }+27 y^{\prime }-27 y = {\mathrm e}^{3 x} \sin \left (x \right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ {\mathrm e}^{3 x} \sin \left (x \right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{{\mathrm e}^{3 x} \cos \left (x \right ), {\mathrm e}^{3 x} \sin \left (x \right )\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{x \,{\mathrm e}^{3 x}, x^{2} {\mathrm e}^{3 x}, {\mathrm e}^{3 x}\} \] 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. \[ y_p = A_{1} {\mathrm e}^{3 x} \cos \left (x \right )+A_{2} {\mathrm e}^{3 x} \sin \left (x \right ) \] The unknowns \(\{A_{1}, A_{2}\}\) 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 \[ A_{1} {\mathrm e}^{3 x} \sin \left (x \right )-A_{2} {\mathrm e}^{3 x} \cos \left (x \right ) = {\mathrm e}^{3 x} \sin \left (x \right ) \] Solving for the unknowns by comparing coefficients results in \[ [A_{1} = 1, A_{2} = 0] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = {\mathrm e}^{3 x} \cos \left (x \right ) \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{3 x} c_{1} +x \,{\mathrm e}^{3 x} c_{2} +x^{2} {\mathrm e}^{3 x} c_{3}\right ) + \left ({\mathrm e}^{3 x} \cos \left (x \right )\right ) \\ \end{align*} Which simplifies to \[ y = {\mathrm e}^{3 x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right )+{\mathrm e}^{3 x} \cos \left (x \right ) \]
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{3 x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right )+{\mathrm e}^{3 x} \cos \left (x \right ) \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{3 x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right )+{\mathrm e}^{3 x} \cos \left (x \right ) \] Verified OK.
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`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 21
dsolve([diff(y(x),x$3)-9*diff(y(x),x$2)+27*diff(y(x),x)-27*y(x)=exp(3*x)*sin(x),exp(3*x)],singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{3 x} \left (\cos \left (x \right )+c_{1} +c_{2} x +x^{2} c_{3} \right ) \]
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 25
DSolve[y'''[x]-9*y''[x]+27*y'[x]-27*y[x]==Exp[3*x]*Sin[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{3 x} (\cos (x)+x (c_3 x+c_2)+c_1) \]