DEtools
reduceOrder
apply the method of reduction of order to an ODE
Calling Sequence
Parameters
Description
Examples
reduceOrder(des, dvar, partsol, solutionForm)
des
-
ordinary differential equation, or its list form
dvar
the dependent variable for an equation
partsol
partial solution, or list of partial solutions
solutionForm
flag to indicate the DE should be solved explicitly
Important: The DEtools[reduceOrder] command has been deprecated. Use the superseding command DEtools[reduce_order] instead.
This routine is used to either a) return an ODE of reduced order or b) solve the ODE explicitly by the method of reduction of order, given a partial (particular) solution of the ODE. Without the optional flag basis, a reduced ODE is returned. If basis appears as the fifth argument, then a list containing the basis of the solution is returned. Note that a solution basis may contain DESol data structures.
des may be input as an explicit ODE, as a list of coefficients (in the case of the ODE being homogeneous), or in the form returned by convertAlg for the non-homogeneous case.
partsol may be a single partial solution, or a list of partial solutions. Note that it is assumed all given partial solutions are correct and valid. When a reduced ODE is to be returned, the order of the resulting ODE will be equal to the order of the original less the number of partial solutions given.
The command with(DEtools,reduceOrder) allows the use of the abbreviated form of this command.
withDEtools:
de≔diffyx,`$`x,3−6diffyx,`$`x,2+11diffyx,x−6yx
de≔ⅆ3ⅆx3yx−6ⅆ2ⅆx2yx+11ⅆⅆxyx−6yx
sol≔expx
sol≔ⅇx
reduceOrderde,yx,sol
ⅆ2ⅆx2yx−3ⅆⅆxyx+2yx
reduceOrderde,yx,sol,basis
ⅇx,ⅇ2x,ⅇ3x2
de2≔24,−50,35,−10,1
sol1≔expx
sol1≔ⅇx
sol2≔exp2x
sol2≔ⅇ2x
reduceOrderde2,yx,sol1
−6,11,−6,1
reduceOrderde2,yx,sol2
2,−1,−2,1
reduceOrderde2,yx,sol1,sol2
2,−3,1
reduceOrderde2,yx,sol1,sol2,basis
ⅇx,ⅇ2x,ⅇ3x2,ⅇ4x6
de3≔x9+x3diffyx,`$`x,3+18x8diffyx,`$`x,2−90xdiffyx,x−3011x6−3yx
de3≔x9+x3ⅆ3ⅆx3yx+18x8ⅆ2ⅆx2yx−90xⅆⅆxyx−3011x6−3yx
sol≔xx6+1
reduceOrderde3,yx,sol
x2ⅆ2ⅆx2yx+3xⅆⅆxyx−90yx
reduceOrderde3,yx,sol,basis
xx6+1,−xx−919191x6+1,xx919191x6+1
See Also
DESol
DEtools[convertAlg]
DEtools[reduce_order]
dsolve
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