a data structure to represent the solution of a differential equation - Maple Help

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DESol - a data structure to represent the solution of a differential equation

Calling Sequence

DESol(expr)

DESol(expr, y)

Parameters

expr

-

differential equation in y

y

-

variable

Description

• 

DESol is a data structure to represent the solution of a differential equation. It is to dsolve as RootOf is to solve.

• 

The DESol data structure makes manipulation of differential equations much easier. Its existence allows many hooks to the rest of Maple, diff, int, evalf, series, simplify, just to name the most prominent.

• 

DESol can be used as an operator (using D), or as an expression (using diff(...,x)). The examples below illustrate some of the operations that can be performed with DESol.

Examples

A simple differential equation.

de1:=DESolDyy,y

de1:=DESolDyy,y

(1)

Verify by differentiating.

Dde1de1

0

(2)

The equivalent equation, as a function:

de2:=DESolⅆⅆxyxyx,yx

de2:=DESolⅆⅆxyxyx,yx

(3)

de1xde2

0

(4)

xde2de2

0

(5)

The unapply command can convert from function to operator.

unapplyde2,xde1

0

(6)

A more complex equation, using initial conditions.

aliasde3=DESolDDyaxDy+y,y:Dx:=1:Da:=0:

DDDde3

aDde3+axaxDde3de3Dde3

(7)

collectDDDDde3,Dde3,factor

axa2x2+3a2Dde3de3a2x2+2a1

(8)

∫de3xⅆx

DESolyxxaⅆⅆxyxa+1+ⅆ2ⅆx2yxa+1,yx

(9)

∫1lnde3xⅆx

DESol2ⅆ2ⅆx2yx2ⅇ1ⅆⅆxyxⅆⅆxyx3ⅆ3ⅆx3yxⅇ1ⅆⅆxyxⅆⅆxyx2+ⅆ2ⅆx2yx2ⅇ1ⅆⅆxyxⅆⅆxyx4+axⅆ2ⅆx2yxⅇ1ⅆⅆxyxⅆⅆxyx2+ⅇ1ⅆⅆxyx,yx

(10)

An example with non-linear DE's:

aliasde5=DESolDDyDy=y:

DDde5Dde5de5

0

(11)

DDDde5

Dde52+de52Dde5

(12)

The large oscillation pendulum equation, with initial conditions:

Dx:='Dx':

de6:=DESolⅆ2ⅆt2xt=gsinxtl,xt,x0=0,Dx0=v0

de6:=DESolⅆ2ⅆt2xtgsinxtl,xt,x0=0,Dx0=v0

(13)

de7:=∫de6ⅆt

de7:=DESolⅆ3ⅆt3xtgsinⅆⅆtxtl,xt

(14)

Verify the derivative.

3t3de7gsintde7l

0

(15)

A complex integration.

de8:=∫1DESolⅆⅆxyxyx,yx+1ⅆx

de8:=lnDESolⅆⅆxyxyx,yxlnDESolⅆⅆxyxyx,yx+1

(16)

normalxde81DESolⅆⅆxyxyx,yx+1

0

(17)

When there are no derivatives, the equation can be written with RootOf or solved:

DESolyx2yx+1,yx

RootOf_Z2_Z+1

(18)

DESolyxx,yx

x

(19)

An example where the equation can be simplified.

de9:=DESolⅆ2ⅆx2yxⅆⅆxyx1,yx

de9:=DESolⅆ2ⅆx2yxⅆⅆxyx1,yx

(20)

xde9

DESolⅆⅆxyxyx1,yx

(21)

de10:=unapplyde9,x

de10:=DESolD2yDy1,y

(22)

Dde10

DESolyDy1,y

(23)

See Also

D, diff, dsolve, int, solve


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