X^3y ''' + 10x^2y '' + 16xy ' − 16y = 0; x, x^−4, x^−4 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−4, x−4 ln x) = ≠ 0 for 0 < x < [infinity].

Form the general solution.

y=

Verifying that a solution satisifes the ODE is a matter of substituting the solution into the ODE. For example,

The Wronskian for these solutions is

which is non-zero for all , so the solutions are indeed linearly independent.

The general solution is then

yess

step-by-step explanation:

idk whyy this is happening, i cant even read my notification bc i keep getting errors

~coco

.25

step-by-step explanation: