SeriesSolutions - Maple Help

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ODE Steps for Series Solutions

 

Overview

Examples

Overview

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This help page gives a few examples of using the command ODESteps to solve ordinary differential equations by means of series expansions.

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See Student[ODEs][ODESteps] for a general description of the command ODESteps and its calling sequence.

Examples

withStudent:-ODEs:

ode1x2diffyx,x,x+xdiffyx,x+5xyx=0

ode1x2ⅆ2ⅆx2yx+xⅆⅆxyx+5xyx=0

(1)

ODEStepsode1

Let's solvex2ⅆ2ⅆx2yx+xⅆⅆxyx+5xyx=0Highest derivative means the order of the ODE is2ⅆ2ⅆx2yxIsolate 2nd derivativeⅆ2ⅆx2yx=5yxxⅆⅆxyxxGroup terms withyxon the lhs of the ODE and the rest on the rhs of the ODE; ODE is linearⅆ2ⅆx2yx+ⅆⅆxyxx+5yxx=0Check to see ifx0=0is a regular singular pointDefine functionsP2x=1x,P3x=5xP2xis analytic atx=0=1P3xis analytic atx=0=0x=0is a regular singular pointCheck to see ifx0=0is a regular singular pointx0=0Multiply by denominatorsⅆ2ⅆx2yxx+5yx+ⅆⅆxyx=0Assume series solution foryxyx=k=0akxk+rRewrite ODE with series expansionsConvertⅆ2ⅆx2yxxto series expansionⅆ2ⅆx2yxx=k=0ak+1k+r+1k+rxk+rConvert5yxto series expansion5yx=k=05akxk+rConvertⅆⅆxyxto series expansionⅆⅆxyx=k=0ak+1k+r+1xk+rRewrite ODE with series expansionsSumak+1k+r+1+ak+1k+r+1k+r+5akxk+r,0..=0a0cannot be 0 by assumption giving the indicial equationr=0Values of r that satisfy the indicial equationr=0Each term must evaluate to 0 giving the recursion relationk+r+12ak+1+5akRecursion relation that defines series solution to ODEak+1=5akk+r+12Solution forr=0yx=k=0akxk,ak+1=5akk+12

(2)

ode2diffyx,x,x+xdiffyx,x+yx=0

ode2ⅆ2ⅆx2yx+xⅆⅆxyx+yx=0

(3)

ODEStepsode2

Let's solveⅆ2ⅆx2yx+xⅆⅆxyx+yx=0Highest derivative means the order of the ODE is2ⅆ2ⅆx2yxAssume series solution foryxyx=k=0akxkRewrite ODE with series expansionsConvertxⅆⅆxyxto series expansionxⅆⅆxyx=k=0akkxkConvertⅆ2ⅆx2yxto series expansionⅆ2ⅆx2yx=k=0ak+2k+2k+1xkConvertyxto series expansionyx=k=0akxkRewrite ODE with series expansionsSumak+akk+ak+2k+2k+1xk,0..=0Each term must evaluate to 0 giving the recursion relationk+1k+2ak+2+ak=0Recursion relation that defines series solution to ODEyx=k=0akxk,ak+2=akk+2

(4)

ode3x2diffyx,x,x+x2diffyx,x+x36yx=0

ode3x2ⅆ2ⅆx2yx+x2ⅆⅆxyx+x36yx=0

(5)

ODEStepsode3

Let's solvex2ⅆ2ⅆx2yx+x2ⅆ