Answer:
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_1......y_2.........y_3|
|y'_1.....y'_2........y'_3|
|y''_1....y''_2.......y''_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|
|1............1........(x - 5)/x + lnx|
|0...........0..............(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.
