Internal problem ID [15323]
Internal file name [OUTPUT/15324_Wednesday_May_08_2024_03_55_47_PM_84433538/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. Superposition principle. Exercises page 137
Problem number: 555.
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, _missing_y]]
\[ \boxed {y^{\prime \prime \prime }-y^{\prime \prime }=1+{\mathrm e}^{x}} \] 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 \[ y^{\prime \prime \prime }-y^{\prime \prime } = 0 \] The characteristic equation is \[ \lambda ^{3}-\lambda ^{2} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 1\\ \lambda _2 &= 0\\ \lambda _3 &= 0 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=x c_{2} +c_{1} +c_{3} {\mathrm e}^{x} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= x \\ y_3 &= {\mathrm e}^{x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }-y^{\prime \prime } = 1+{\mathrm e}^{x} \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ 1+{\mathrm e}^{x} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{1\}, \{{\mathrm e}^{x}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{1, x, {\mathrm e}^{x}\} \] Since \(1\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x\}, \{{\mathrm e}^{x}\}] \] Since \(x\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x^{2}\}, \{{\mathrm e}^{x}\}] \] Since \({\mathrm e}^{x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x^{2}\}, \{{\mathrm e}^{x} x\}] \] 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. \[ y_p = A_{1} x^{2}+A_{2} {\mathrm e}^{x} x \] 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 \[ A_{2} {\mathrm e}^{x}-2 A_{1} = 1+{\mathrm e}^{x} \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = -{\frac {1}{2}}, A_{2} = 1\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = -\frac {x^{2}}{2}+{\mathrm e}^{x} x \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (x c_{2} +c_{1} +c_{3} {\mathrm e}^{x}\right ) + \left (-\frac {x^{2}}{2}+{\mathrm e}^{x} x\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= x c_{2} +c_{1} +c_{3} {\mathrm e}^{x}-\frac {x^{2}}{2}+{\mathrm e}^{x} x \\ \end{align*}
Verification of solutions
\[ y = x c_{2} +c_{1} +c_{3} {\mathrm e}^{x}-\frac {x^{2}}{2}+{\mathrm e}^{x} x \] Verified OK.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable -> Calling odsolve with the ODE`, diff(_b(_a), _a) = _b(_a)+1+exp(_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: 21
dsolve(diff(y(x),x$3)-diff(y(x),x$2)=1+exp(x),y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{1} +x -2\right ) {\mathrm e}^{x}-\frac {x^{2}}{2}+c_{2} x +c_{3} \]
✓ Solution by Mathematica
Time used: 0.115 (sec). Leaf size: 28
DSolve[y'''[x]-y''[x]==1+Exp[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {x^2}{2}+c_3 x+e^x (x-2+c_1)+c_2 \]