Consider the differential equation x3y ''' + 12x2y '' + 25xy ' − 25y = 0; x, x−5, x−5 ln(x), (0, [infinity]). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. The functions satisfy the differential equation and are linearly independent since W(x, x−5, x−5 ln(x))

The set of solutions are independent, and therefore form a fundamental set of solution.

Step-by-step explanation:

We want to verify that the functions x, x - 5, (x - 5)lnx for a fundamental set of solutions to the differential equation

x³y''' + 12x²y'' + 25xy' - 25y = 0

On the interval (0, infinity).

By definition, the Wronskian of y_1, y_2, y_3 is given as the determinant:

W(y_1, y_2, y_3) =

|y_1y_2y_3|

|y'_1y'_2y'_3|

|y''_1y''_2y''_3|

If W(y_1, y_2, y_3) ≠ 0, the functions are independent.

If the funtions form a fundamental set of solutions to the differential equation, then they are independent, otherwise, the are dependent, and hence, do not form a set of fundamental solution.

Now, let us check if the given functions form a set of fundamental solution.

W(x, x - 5, x - 5 ln(x)) =

|x(x - 5)(x - 5)lnx|

|11(x - 5)/x + lnx|

|00(5 + x)/x²|

= x((5 + x)/x² - 0) - (x - 5)((x + 5)/x² - 0) + (x - 5)lnx(0 - 0)

= (x + 5)/x - (x - 5)(x + 5)/x²

= (x + 5)/x - (x² - 25)/x²

= (5x + x² - x² - 25)/x²

= (5x - 25)/x²

= 5(x - 5)/x²

≠0

Hence, the set of solutions are independent, and therefore form a fundamental set of solution.

median is the value separating the higher half of a data sample, a population or probablity distribution from the lower half.

step-by-step explanation:

say you have 3,8,7,3,1,9,8 order them from lowest to highest

if orderd it would be 1,3,3,7,8,8,9 now you take away from the lowest and the highest until you end up in the middle.

so you would remove the 1 and 9 leaving you with 5 digits remaining

when you've reached the middle that would be your answer so in this case the answer is 7

if there were two numbers left say 7 and 5 you'd and them together so 12 then divide by 2 witch would equal 6 so the answer would be 6

good luck on that project hoped i you at all