Series solutions to differential equations, Mathematics

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Before searching at series solutions to a differential equation we will initially require to do a cursory review of power series. So, a power series is a series in the form,

1570_SERIES SOLUTIONS TO DIFFERENTIAL EQUATIONS.png ........................(1)

Here, x0 and an are numbers. We can notice from that as a power series is a function of x. The function notation is not all the time contained, but sometimes this is so we place this in the definition above.

Before proceeding along with our review we must probably first recall just what series actually are. Recall that series are actually just summations. Then one method to write our power series is,

2147_SERIES SOLUTIONS TO DIFFERENTIAL EQUATIONS1.png

= a0 + a1 (x - x0) + a2 (x - x0)2 + a3 (x - x0)3+  ..............            (2)

Notice finely that if we required to for some purpose we could all the time write the power series as,

1699_SERIES SOLUTIONS TO DIFFERENTIAL EQUATIONS2.png

= a0 + a1 (x - x0) + a2 (x - x0)2 + a3 (x - x0)3+..............             

581_SERIES SOLUTIONS TO DIFFERENTIAL EQUATIONS3.png

All which we're doing now there is noticing that if we avoid the first term (consequent to n = 0) the remains is just a series which starts at n = 1. While we do this we say which we've stripped out the n = 0, or first term. We don't require stopping at the first term either. If we strip out the initially three terms we'll find,

1620_SERIES SOLUTIONS TO DIFFERENTIAL EQUATIONS4.png

There are times while we'll need to do this so ensure that you can do it.

Here, as power series are functions of x and we know that not each series will actually exist, this then makes sense to ask if a power series will exist for all x. This question is answered by searching at the convergence of the power series. We say as a power series converges for x = c whether the series, converges.

1155_SERIES SOLUTIONS TO DIFFERENTIAL EQUATIONS5.png

Recall here that series which will converge if the restrict of partial sums,

1819_SERIES SOLUTIONS TO DIFFERENTIAL EQUATIONS6.png

exists and it is finite.  Conversely, a power series will converge for x=c if

1153_SERIES SOLUTIONS TO DIFFERENTIAL EQUATIONS7.png

above is a finite number.

Remember that a power series will all the time converge if x = x0. During this case the power series will become ∞;

217_SERIES SOLUTIONS TO DIFFERENTIAL EQUATIONS8.png

With this we here know that power series are guaranteed to exist for in any case one value of x. We have the subsequent fact regarding to the convergence of a power series.


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