Verify that the function(s) solve the following differential equations (des): a) y' = -5y; y = 3e-5x b) y' = cos(3x); y = į sin(3x) + 7 c) y' = 2y; y = ce2x , where c is any real number. d) y" + y' – 6y = 0 ; yı = (2x, y2 = (–3x e) y" + 16y = 0; yı = cos(4x), y2 = sin(4x)

In the step-by-step explanation, the verifications are made.

Step-by-step explanation:

a)

This one can be solved by the variable separation method

The value of C is the value of y when x = 0. If , then we have the following solution:

b)

This one can also be solved by the variable separation method

K is also the value of y, when x = 0. So, if , we have the following solution.

c)

Another one that can be solved by the variable separation method

C is any real number depending on the initial conditions.

d)

Here, the solution depends on the roots of the following equation:

or .

So the solution is

The values of depends on the initial conditions.

e)

Again, we find the roots of the following equation:

So we have the following solution

The values of depends on the initial conditions.