Find a power series solution of the differential equation y" + 4xy = 0 about the ordinary point x = 0.

Step-by-step explanation:

Assume that the problem has a solution of the form:

then calculate the derivative:

and the second derivative:

In these sums the subindex indicates the first non cero term.

Making the substitution in the original equation:

Now look at the subindex of the sum: they do not match; you need to make them match:

Now they match and they could be joined together on the same sum symbol, however the x term doesn't have the same exponent (after multiplying the x from the original formula with the one inside the sum). Therefore you need to expand the second derivative term and displace the other one (as we did in the previous step) to match them both:

Now you can join them together:

If the solution is valid for all x, then all terms must be independantly 0. therefore:

Now you need to use that recurrent function to calculate the coefficients:

you found that for each n divisible by 3, therefore for each the coefficient is 0:

now look at the terms , they are all recurrent to c_0, therefore you can wirte a rule for them:

Finally, look at the terms , they recurr to c_1, therefore:

So you have two constants that are independant, c_0 and c_1.

Therefore the solution must be writen in terms of these two arbitrary constants:

the smallest possible measurement of the width is 14 feet

step-by-step explanation:

let's assume length of parallelogram is 'l'

width of parallelogram is 'w'

the perimeter of a parallelogram must be no less than 40 feet

so, we get

we are given

length of parallelogram is 6 feet

so,

l=6

we can plug it

now, we can solve for w

subtract both sides by 12

divide both sides by 2

and we get

so,

the smallest possible measurement of the width is 14 feet

false

step-by-step explanation: