Internal problem ID [9870]
Internal file name [OUTPUT/8817_Monday_June_06_2022_05_33_08_AM_38583927/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1548.
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]]
\[ \boxed {4 y^{\prime \prime \prime \prime }-12 y^{\prime \prime \prime }+11 y^{\prime \prime }-3 y^{\prime }=4 \cos \left (x \right )} \] 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 \[ 4 y^{\prime \prime \prime \prime }-12 y^{\prime \prime \prime }+11 y^{\prime \prime }-3 y^{\prime } = 0 \] The characteristic equation is \[ 4 \lambda ^{4}-12 \lambda ^{3}+11 \lambda ^{2}-3 \lambda = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 1\\ \lambda _3 &= {\frac {1}{2}}\\ \lambda _4 &= {\frac {3}{2}} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} +{\mathrm e}^{\frac {x}{2}} c_{2} +{\mathrm e}^{x} c_{3} +{\mathrm e}^{\frac {3 x}{2}} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= {\mathrm e}^{\frac {x}{2}} \\ y_3 &= {\mathrm e}^{x} \\ y_4 &= {\mathrm e}^{\frac {3 x}{2}} \\ \end{align*} Now the particular solution to the given ODE is found \[ 4 y^{\prime \prime \prime \prime }-12 y^{\prime \prime \prime }+11 y^{\prime \prime }-3 y^{\prime } = 4 \cos \left (x \right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ 4 \cos \left (x \right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{\cos \left (x \right ), \sin \left (x \right )\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \left \{1, {\mathrm e}^{x}, {\mathrm e}^{\frac {x}{2}}, {\mathrm e}^{\frac {3 x}{2}}\right \} \] 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. \[ y_p = A_{1} \cos \left (x \right )+A_{2} \sin \left (x \right ) \] 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 \[ -7 A_{1} \cos \left (x \right )-7 A_{2} \sin \left (x \right )-9 A_{1} \sin \left (x \right )+9 A_{2} \cos \left (x \right ) = 4 \cos \left (x \right ) \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = -{\frac {14}{65}}, A_{2} = {\frac {18}{65}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = -\frac {14 \cos \left (x \right )}{65}+\frac {18 \sin \left (x \right )}{65} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} +{\mathrm e}^{\frac {x}{2}} c_{2} +{\mathrm e}^{x} c_{3} +{\mathrm e}^{\frac {3 x}{2}} c_{4}\right ) + \left (-\frac {14 \cos \left (x \right )}{65}+\frac {18 \sin \left (x \right )}{65}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} +{\mathrm e}^{\frac {x}{2}} c_{2} +{\mathrm e}^{x} c_{3} +{\mathrm e}^{\frac {3 x}{2}} c_{4} -\frac {14 \cos \left (x \right )}{65}+\frac {18 \sin \left (x \right )}{65} \\ \end{align*}
Verification of solutions
\[ y = c_{1} +{\mathrm e}^{\frac {x}{2}} c_{2} +{\mathrm e}^{x} c_{3} +{\mathrm e}^{\frac {3 x}{2}} c_{4} -\frac {14 \cos \left (x \right )}{65}+\frac {18 \sin \left (x \right )}{65} \] 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(diff(_b(_a), _a), _a), _a) = 3*(diff(diff(_b(_a), _a), _a))-(11/4)*(diff(_b(_a), _a))+(3 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 <- differential order: 4; linear nonhomogeneous with symmetry [0,1] successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 32
dsolve(4*diff(diff(diff(diff(y(x),x),x),x),x)-12*diff(diff(diff(y(x),x),x),x)+11*diff(diff(y(x),x),x)-3*diff(y(x),x)-4*cos(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{x}+2 c_{2} {\mathrm e}^{\frac {x}{2}}+\frac {2 c_{3} {\mathrm e}^{\frac {3 x}{2}}}{3}+\frac {18 \sin \left (x \right )}{65}-\frac {14 \cos \left (x \right )}{65}+c_{4} \]
✓ Solution by Mathematica
Time used: 0.112 (sec). Leaf size: 50
DSolve[-4*Cos[x] - 3*y'[x] + 11*y''[x] - 12*Derivative[3][y][x] + 4*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {18 \sin (x)}{65}-\frac {14 \cos (x)}{65}+2 c_1 e^{x/2}+\frac {2}{3} c_2 e^{3 x/2}+c_3 e^x+c_4 \]