Like Tony Stark
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- Apr 19, 2020
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Hi
Given any linear homogenous ODE, how can I tell what's the behavior of its solutions?
It is easy if we consider an equation with constant coefficients because its behavior is "controlled" by exponential and trigonometric functions. But what if we have an equation which must be solved with Frobenius method, where the solutions are power series (analytic solutions)?
Consider the following example:
x²y''+xy'-xy=0
This last equation gives a double root for the inditial equation: r=0
What happens when x approaches 0? As the solution is a power series, if x=0, then y=0. So is this the right way to say that Y approaches zero as X approaches zero?
Given any linear homogenous ODE, how can I tell what's the behavior of its solutions?
It is easy if we consider an equation with constant coefficients because its behavior is "controlled" by exponential and trigonometric functions. But what if we have an equation which must be solved with Frobenius method, where the solutions are power series (analytic solutions)?
Consider the following example:
x²y''+xy'-xy=0
This last equation gives a double root for the inditial equation: r=0
What happens when x approaches 0? As the solution is a power series, if x=0, then y=0. So is this the right way to say that Y approaches zero as X approaches zero?