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(10x^2D^2-20xD+22.4I)y=0...no idea to find it by using euler-method...help..sos

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- Thread starter xw0927
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In summary, the conversation discusses different methods for finding the general solution to the given equation and mentions the use of Euler's name in various contexts. The equation can be solved by trying a solution of the form y= x^r, making a change of variable, or using Euler's numerical method for first order equations. The final solution given is C1x^1.4 + C2x^1.6 and there are no complex solutions. The conversation also mentions the Euler-Cauchy equation.

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(10x^2D^2-20xD+22.4I)y=0...no idea to find it by using euler-method...help..sos

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- #2

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That

[itex](10r(r-1)+ 20r+ 22.4)x^r= 0[/itex] and, since [itex]x^r[/itex] is not always 0,

[itex]10r(r- 1)+ 20r+ 22.4= 10r^2+ 10r+ 22.4= 0[/itex]

That will have complex solutions so you would have to interpret [itex]x^{a+ bi}[/itex] in terms of sine and cosine.

Another way of doing that would be to make the change of variable x= ln(t) which converts the "Euler-type" equation to an equation with constant coefficents having the same characteristic equation.

Or, since you say "Euler

Let y= Dy so that [itex]D^2 y= Du[/itex] any your equation becomes

[itex]x^2Du- 20xDy+ 22.4y= 0[/itex]

and [itex]Dy= u[/itex]

You can set up an "double" Euler-method solver that, given the values of u and y at a each step, solves Dy= u for y and then puts both of those values into the other equation to solve for y.

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