Internal problem ID [2225]
Internal file name [OUTPUT/2225_Monday_February_26_2024_09_18_32_AM_17887916/index.tex]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 23, page 106
Problem number: 24.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+9 y^{\prime \prime }=\sin \left (3 x \right )+x \,{\mathrm e}^{x}} \] 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 \[ y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+9 y^{\prime \prime } = 0 \] The characteristic equation is \[ \lambda ^{4}-6 \lambda ^{3}+9 \lambda ^{2} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 0\\ \lambda _3 &= 3\\ \lambda _4 &= 3 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{4} x \,{\mathrm e}^{3 x}+{\mathrm e}^{3 x} c_{3} +c_{2} x +c_{1} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= x \\ y_3 &= {\mathrm e}^{3 x} \\ y_4 &= x \,{\mathrm e}^{3 x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+9 y^{\prime \prime } = \sin \left (3 x \right )+x \,{\mathrm e}^{x} \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ \sin \left (3 x \right )+x \,{\mathrm e}^{x} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{x \,{\mathrm e}^{x}, {\mathrm e}^{x}\}, \{\cos \left (3 x \right ), \sin \left (3 x \right )\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{1, x, x \,{\mathrm e}^{3 x}, {\mathrm e}^{3 x}\} \] 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. \[ y_p = A_{1} x \,{\mathrm e}^{x}+A_{2} {\mathrm e}^{x}+A_{3} \cos \left (3 x \right )+A_{4} \sin \left (3 x \right ) \] 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 \[ 4 A_{1} {\mathrm e}^{x}+4 A_{1} x \,{\mathrm e}^{x}+4 A_{2} {\mathrm e}^{x}-162 A_{3} \sin \left (3 x \right )+162 A_{4} \cos \left (3 x \right ) = \sin \left (3 x \right )+x \,{\mathrm e}^{x} \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = {\frac {1}{4}}, A_{2} = -{\frac {1}{4}}, A_{3} = -{\frac {1}{162}}, A_{4} = 0\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {x \,{\mathrm e}^{x}}{4}-\frac {{\mathrm e}^{x}}{4}-\frac {\cos \left (3 x \right )}{162} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{4} x \,{\mathrm e}^{3 x}+{\mathrm e}^{3 x} c_{3} +c_{2} x +c_{1}\right ) + \left (\frac {x \,{\mathrm e}^{x}}{4}-\frac {{\mathrm e}^{x}}{4}-\frac {\cos \left (3 x \right )}{162}\right ) \\ \end{align*} Which simplifies to \[ y = \left (c_{4} x +c_{3} \right ) {\mathrm e}^{3 x}+c_{2} x +c_{1} +\frac {x \,{\mathrm e}^{x}}{4}-\frac {{\mathrm e}^{x}}{4}-\frac {\cos \left (3 x \right )}{162} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \left (c_{4} x +c_{3} \right ) {\mathrm e}^{3 x}+c_{2} x +c_{1} +\frac {x \,{\mathrm e}^{x}}{4}-\frac {{\mathrm e}^{x}}{4}-\frac {\cos \left (3 x \right )}{162} \\ \end{align*}
Verification of solutions
\[ y = \left (c_{4} x +c_{3} \right ) {\mathrm e}^{3 x}+c_{2} x +c_{1} +\frac {x \,{\mathrm e}^{x}}{4}-\frac {{\mathrm e}^{x}}{4}-\frac {\cos \left (3 x \right )}{162} \] 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*exp(_a)+sin(3*_a)-9*_b(_a)+6*(diff(_b(_a), _a)), _b(_a)` *** Sub 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: 0.015 (sec). Leaf size: 39
dsolve(diff(y(x),x$4)-6*diff(y(x),x$3)+9*diff(y(x),x$2)=sin(3*x)+x*exp(x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (3 c_{1} x -2 c_{1} +3 c_{2} \right ) {\mathrm e}^{3 x}}{27}-\frac {\cos \left (3 x \right )}{162}+\frac {\left (x -1\right ) {\mathrm e}^{x}}{4}+c_{3} x +c_{4} \]
✓ Solution by Mathematica
Time used: 1.069 (sec). Leaf size: 52
DSolve[y''''[x]-6*y'''[x]+9*y''[x]==Sin[3*x]+x*Exp[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{4} e^x (x-1)-\frac {1}{162} \cos (3 x)+\frac {1}{27} e^{3 x} (c_2 (3 x-2)+3 c_1)+c_4 x+c_3 \]