Internal problem ID [15312]
Internal file name [OUTPUT/15313_Wednesday_May_08_2024_03_55_29_PM_46704334/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 15.3 Nonhomogeneous linear equations with
constant coefficients. Trial and error method. Exercises page 132
Problem number: 543.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }+y^{\prime \prime }=x^{2}+x} \] 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 \[ y^{\prime \prime \prime \prime }+y^{\prime \prime } = 0 \] The characteristic equation is \[ \lambda ^{4}+\lambda ^{2} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 0\\ \lambda _3 &= i\\ \lambda _4 &= -i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=x c_{2} +c_{1} +{\mathrm e}^{i x} c_{3} +{\mathrm e}^{-i x} c_{4} \] 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 x} \\ y_4 &= {\mathrm e}^{-i x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }+y^{\prime \prime } = x^{2}+x \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ x^{2}+x \] 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 \[ \{1, x, {\mathrm e}^{i x}, {\mathrm e}^{-i x}\} \] 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 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. \[ y_p = A_{3} x^{4}+A_{2} x^{3}+A_{1} x^{2} \] 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 \[ 12 x^{2} A_{3}+6 x A_{2}+2 A_{1}+24 A_{3} = x^{2}+x \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = -1, A_{2} = {\frac {1}{6}}, A_{3} = {\frac {1}{12}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {1}{12} x^{4}+\frac {1}{6} x^{3}-x^{2} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (x c_{2} +c_{1} +{\mathrm e}^{i x} c_{3} +{\mathrm e}^{-i x} c_{4}\right ) + \left (\frac {1}{12} x^{4}+\frac {1}{6} x^{3}-x^{2}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= x c_{2} +c_{1} +{\mathrm e}^{i x} c_{3} +{\mathrm e}^{-i x} c_{4} +\frac {x^{4}}{12}+\frac {x^{3}}{6}-x^{2} \\ \end{align*}
Verification of solutions
\[ y = x c_{2} +c_{1} +{\mathrm e}^{i x} c_{3} +{\mathrm e}^{-i x} c_{4} +\frac {x^{4}}{12}+\frac {x^{3}}{6}-x^{2} \] 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) = _a^2-_b(_a)+_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: 0.0 (sec). Leaf size: 34
dsolve(diff(y(x),x$4)+diff(y(x),x$2)=x^2+x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{3}}{6}-x^{2}+\frac {x^{4}}{12}-\cos \left (x \right ) c_{1} -\sin \left (x \right ) c_{2} +c_{3} x +c_{4} \]
✓ Solution by Mathematica
Time used: 0.073 (sec). Leaf size: 43
DSolve[y''''[x]+y''[x]==x^2+x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^4}{12}+\frac {x^3}{6}-x^2+c_4 x-c_1 \cos (x)-c_2 \sin (x)+c_3 \]