Internal
problem
ID
[8577]
Book
:
Second
order
enumerated
odes
Section
:
section
2
Problem
number
:
42
Date
solved
:
Sunday, November 10, 2024 at 04:05:27 AM
CAS
classification
:
[[_high_order, _linear, _nonhomogeneous]]
Solve
Time used: 0.161 (sec)
The characteristic equation is
The roots of the above equation are
Therefore the homogeneous solution is
The fundamental set of solutions for the homogeneous solution are the following
This is higher order nonhomogeneous ODE. Let the solution be
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
Now the particular solution to the given ODE is found
The particular solution is now found using the method of undetermined coefficients.
Looking at the RHS of the ode, which is
Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is
While the set of the basis functions for the homogeneous solution found earlier is
Since \({\mathrm e}^{-2 x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes
Since \({\mathrm e}^{x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes
Since \(x \,{\mathrm e}^{x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes
Since \(x^{2} {\mathrm e}^{x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes
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.
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
Solving for the unknowns by comparing coefficients results in
Substituting the above back in the above trial solution \(y_p\), gives the particular solution
Therefore the general solution is
`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] trying high order linear exact nonhomogeneous trying differential order: 4; missing the dependent variable checking if the LODE has constant coefficients <- constant coefficients successful`
Solving time : 0.010
(sec)
Leaf size : 52
dsolve(diff(diff(diff(diff(y(x),x),x),x),x)-diff(diff(diff(y(x),x),x),x)-3*diff(diff(y(x),x),x)+5*diff(y(x),x)-2*y(x) = exp(x)*x+3*exp(-2*x), y(x),singsol=all)
Solving time : 0.408
(sec)
Leaf size : 64
DSolve[{D[y[x],{x,4}]-D[y[x],{x,3}]-3*D[y[x],{x,2}]+5*D[y[x],x]-2*y[x]==x*Exp[x]+3*Exp[-2*x],{}}, y[x],x,IncludeSingularSolutions->True]