Internal problem ID [15317]
Internal file name [OUTPUT/15318_Wednesday_May_08_2024_03_55_39_PM_18569722/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: 548.
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]]
\[ \boxed {y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y={\mathrm e}^{x} \cos \left (2 x \right )} \] 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 \[ y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y = 0 \] The characteristic equation is \[ \lambda ^{3}-3 \lambda ^{2}+3 \lambda -1 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 1\\ \lambda _2 &= 1\\ \lambda _3 &= 1 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{x}+x \,{\mathrm e}^{x} c_{2} +x^{2} {\mathrm e}^{x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{x} \\ y_2 &= {\mathrm e}^{x} x \\ y_3 &= {\mathrm e}^{x} x^{2} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y = {\mathrm e}^{x} \cos \left (2 x \right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ {\mathrm e}^{x} \cos \left (2 x \right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{{\mathrm e}^{x} \cos \left (2 x \right ), {\mathrm e}^{x} \sin \left (2 x \right )\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{{\mathrm e}^{x} x, {\mathrm e}^{x} x^{2}, {\mathrm e}^{x}\} \] 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. \[ y_p = A_{1} {\mathrm e}^{x} \cos \left (2 x \right )+A_{2} {\mathrm e}^{x} \sin \left (2 x \right ) \] 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 \[ 8 A_{1} {\mathrm e}^{x} \sin \left (2 x \right )-8 A_{2} {\mathrm e}^{x} \cos \left (2 x \right ) = {\mathrm e}^{x} \cos \left (2 x \right ) \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = 0, A_{2} = -{\frac {1}{8}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = -\frac {{\mathrm e}^{x} \sin \left (2 x \right )}{8} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} {\mathrm e}^{x}+x \,{\mathrm e}^{x} c_{2} +x^{2} {\mathrm e}^{x} c_{3}\right ) + \left (-\frac {{\mathrm e}^{x} \sin \left (2 x \right )}{8}\right ) \\ \end{align*} Which simplifies to \[ y = {\mathrm e}^{x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right )-\frac {{\mathrm e}^{x} \sin \left (2 x \right )}{8} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right )-\frac {{\mathrm e}^{x} \sin \left (2 x \right )}{8} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right )-\frac {{\mathrm e}^{x} \sin \left (2 x \right )}{8} \] 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: 0.016 (sec). Leaf size: 29
dsolve(diff(y(x),x$3)-3*diff(y(x),x$2)+3*diff(y(x),x)-y(x)=exp(x)*cos(2*x),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {{\mathrm e}^{x} \left (-8 c_{3} x^{2}-8 c_{2} x +\sin \left (2 x \right )-8 c_{1} -2 x \right )}{8} \]
✓ Solution by Mathematica
Time used: 0.069 (sec). Leaf size: 33
DSolve[y'''[x]-3*y''[x]+3*y'[x]-y[x]==Exp[x]*Cos[2*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{8} e^x (-\sin (2 x)+8 (x (c_3 x+c_2)+c_1)) \]