A. seek power series solutions of the given differential equation about the given point x0; find the recurrence relation that the coefficients must satisfy.

b. find the first four nonzero terms in each of two solutions y1 and y2 (unless the series terminates sooner).

c. by evaluating the wronskian w[y1, y2](x0), show that y1 and y2 form a fundamental set of solutions.

d. if possible, find the general term in each solution.

y" + 3y' = 0, x0 = 0

We want to find a solution such that

With these conditions, and .

Substituting the series into the ODE gives

so that the coefficients of satisfy

for , or

for . Notice that this implies a dependency of all beyond on , while takes on whatever initial value is given. In particular,

and so on up to

So we can extract two fundamental solutions such that , where

Recall that

which tells us

but is a constant solution and already accounts for the constant term in , and can be reduced to a simpler constant , leaving us with

The Wronskian is

so the two solutions are indeed independent as long as neither initial value is 0.

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step-by-step explanation:

hi there!

first we need to find out how much they are adding to each number to make the next number.

ex: 4 + 4 = 8, 8 + 8 = 16, and so on.

so continuing with 32,

32 + 32 = 64

64 + 64 = 128

128 + 128 = 256

256 + 256 = 512

4, 8, 16, 32, 64, 128, 256, 512

the 8th term of the sequence will be 512

hope this ! ~