Internal problem ID [3262]
Internal file name [OUTPUT/2754_Sunday_June_05_2022_08_40_05_AM_24257591/index.tex]
Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 4. Linear Differential Equations. Page 183
Problem number: 21.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime }=7 x -3 \cos \left (x \right )} \] 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 \[ y^{\prime \prime \prime \prime }+2 y^{\prime \prime } = 0 \] The characteristic equation is \[ \lambda ^{4}+2 \lambda ^{2} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 0\\ \lambda _3 &= i \sqrt {2}\\ \lambda _4 &= -i \sqrt {2} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{2} x +c_{1} +{\mathrm e}^{-i \sqrt {2}\, x} c_{3} +{\mathrm e}^{i \sqrt {2}\, x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= x \\ y_3 &= {\mathrm e}^{-i \sqrt {2}\, x} \\ y_4 &= {\mathrm e}^{i \sqrt {2}\, x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }+2 y^{\prime \prime } = 7 x -3 \cos \left (x \right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ 7 x -3 \cos \left (x \right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{1, x\}, \{\cos \left (x \right ), \sin \left (x \right )\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \left \{1, x, {\mathrm e}^{i \sqrt {2}\, x}, {\mathrm e}^{-i \sqrt {2}\, x}\right \} \] Since \(1\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x, x^{2}\}, \{\cos \left (x \right ), \sin \left (x \right )\}] \] Since \(x\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x^{2}, x^{3}\}, \{\cos \left (x \right ), \sin \left (x \right )\}] \] 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. \[ y_p = A_{2} x^{3}+A_{1} x^{2}+A_{3} \cos \left (x \right )+A_{4} \sin \left (x \right ) \] The unknowns \(\{A_{1}, A_{2}, A_{3}, A_{4}\}\) 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 \[ -A_{3} \cos \left (x \right )-A_{4} \sin \left (x \right )+12 A_{2} x +4 A_{1} = 7 x -3 \cos \left (x \right ) \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = 0, A_{2} = {\frac {7}{12}}, A_{3} = 3, A_{4} = 0\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {7 x^{3}}{12}+3 \cos \left (x \right ) \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{2} x +c_{1} +{\mathrm e}^{-i \sqrt {2}\, x} c_{3} +{\mathrm e}^{i \sqrt {2}\, x} c_{4}\right ) + \left (\frac {7 x^{3}}{12}+3 \cos \left (x \right )\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} x +c_{1} +{\mathrm e}^{-i \sqrt {2}\, x} c_{3} +{\mathrm e}^{i \sqrt {2}\, x} c_{4} +\frac {7 x^{3}}{12}+3 \cos \left (x \right ) \\ \end{align*}
Verification of solutions
\[ y = c_{2} x +c_{1} +{\mathrm e}^{-i \sqrt {2}\, x} c_{3} +{\mathrm e}^{i \sqrt {2}\, x} c_{4} +\frac {7 x^{3}}{12}+3 \cos \left (x \right ) \] 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: 4; linear nonhomogeneous with symmetry [0,1] -> Calling odsolve with the ODE`, diff(diff(_b(_a), _a), _a) = -2*_b(_a)+7*_a-3*cos(_a), _b(_a)` *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 2; linear nonhomogeneous with symmetry [0,1] trying a double symmetry of the form [xi=0, eta=F(x)] -> Try solving first the homogeneous part of the ODE checking if the LODE has constant coefficients <- constant coefficients successful <- solving first the homogeneous part of the ODE successful <- differential order: 4; linear nonhomogeneous with symmetry [0,1] successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 36
dsolve(diff(y(x),x$4)+2*diff(y(x),x$2)=7*x-3*cos(x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {7 x^{3}}{12}-\frac {\cos \left (\sqrt {2}\, x \right ) c_{1}}{2}-\frac {c_{2} \sin \left (\sqrt {2}\, x \right )}{2}+3 \cos \left (x \right )+c_{3} x +c_{4} \]
✓ Solution by Mathematica
Time used: 0.603 (sec). Leaf size: 51
DSolve[y''''[x]+2*y''[x]==7*x-3*Cos[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {7 x^3}{12}+3 \cos (x)+c_4 x-\frac {1}{2} c_1 \cos \left (\sqrt {2} x\right )-\frac {1}{2} c_2 \sin \left (\sqrt {2} x\right )+c_3 \]