Internal
problem
ID
[8254]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
2.0
Problem
number
:
23
Date
solved
:
Sunday, November 10, 2024 at 03:30:00 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
Solve
Time used: 0.071 (sec)
This is Airy ODE. It has the general form
Where in this case
Therefore the solution to the homogeneous Airy ODE becomes
Since this is inhomogeneous Airy ODE, then we need to find the particular solution. 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 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.
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
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
Will add steps showing solving for IC soon.
Summary of solutions found
`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 checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful <- solving first the homogeneous part of the ODE successful`
Solving time : 0.021
(sec)
Leaf size : 31
dsolve(diff(diff(y(x),x),x)-4*diff(y(x),x)-x*y(x)-x^3+2 = 0, y(x),singsol=all)
Solving time : 2.634
(sec)
Leaf size : 89
DSolve[{D[y[x],{x,2}]-4*D[y[x],x]-x*y[x]-x^3+2==0,{}}, y[x],x,IncludeSingularSolutions->True]