hbomb said:I need some help with power series.
I can't remember how to find a power series center around a point.
example question:
y"-xy'-y=0, x=1
I don't how to start this.
A power series is an infinite series of the form ∑n=0^∞ an(x-c)n, where an are the coefficients, x is the variable, and c is the center of the series. It is used to represent a function as a sum of terms with increasing powers of x.
By substituting the power series into the differential equation, we can find the recurrence relation between the coefficients. Solving this recurrence relation will give us the coefficients, which can be used to construct the power series solution to the differential equation.
Power series solutions can be used to approximate solutions to differential equations with high accuracy. They also provide a systematic way of finding solutions to differential equations, and can be used in cases where other methods may not be applicable.
Power series solutions may not converge for all values of x, and in some cases, may only converge for a limited interval. Additionally, finding the recurrence relation and solving it to obtain the coefficients can be a tedious and time-consuming process.
Yes, power series can be used to solve higher order differential equations. However, as the order of the differential equation increases, the recurrence relation becomes more complex and the solution may only converge for a smaller interval or may not converge at all.