Can Variable Coefficients and Nonhomogeneous Terms Be Solved Using Power Series?

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In summary, the speaker is asking for help with a differential equation with variable coefficients and a nonhomogeneous term. They have tried using a power series solution but are unsure which method to use for the nonhomogeneous term. They also mention the need for a general solution with complementary and particular parts. A suggestion is made to rewrite the equation in terms of a negative power or to multiply by x^2 and include the constant term in the series.
  • #1
rumy
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hello everybody,
i have a different DE question, actually i have searced it in the net but didn't find the answer.
what if there is a DE with variable coefficients (needed power series sol'n)
but a NONHOMOGENEOUS one? (actually it's kinda urgent, have only one day)
for example: x^2 y'' + xy' + (x+1)y = 1/x^2
my sol'n:
actually i can't think of anything, if the righthand side was 0 it could be solved by
y=SUM( an * x^(n+r) ) then apply the well known solution (the long one, can't be written here, it is very long)
but i don't know even which method to apply to solve a nonhomo
by the way if the general solution i need is y = yc + yp
yc: y complementary
yp: y particular
i need yp (don't know which method to use)
 
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  • #2
Write the righthand side as x-2 and fit it into your series!

Or, if you don't like that negative power, multiply the entire equation by x^2 and write it as x^4 y"+ x^3y'+ x^2(x+1)y= 1. Include the 1 in your series for y.
 

FAQ: Can Variable Coefficients and Nonhomogeneous Terms Be Solved Using Power Series?

What is the difference between differential equations and regular equations?

Differential equations involve variables that change over time, while regular equations involve only fixed quantities. This makes differential equations more useful for modeling dynamic systems.

How are differential equations used in scientific research?

Differential equations are used to describe and model natural phenomena in fields such as physics, chemistry, biology, and engineering. They allow scientists to make predictions and understand the behavior of complex systems.

What are the different types of differential equations?

The main types of differential equations are ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve a single independent variable, while PDEs involve multiple independent variables.

Can differential equations be solved analytically or numerically?

Some simple differential equations can be solved analytically by finding an explicit formula for the solution. However, many more complex equations must be solved numerically using computer algorithms.

Are there real-world applications of differential equations?

Yes, differential equations are used in a wide range of real-world applications, including modeling population growth, predicting weather patterns, designing electrical circuits, and simulating chemical reactions.

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