Solve the initial-value problem: y" - 4y' + 8y = 0, y(0) = 1, y'(0) = 2.

The solution of the problem is

Step-by-step explanation:

First we will write the characteristic equation which is

Now, we will solve this quadratic equation using the general formula.

Given a quadratic equation of the form, , then

From the general formula,

or

From the characteristic equation, and

Hence,

or

or

or

or

That is, =   ±

Then, and

These are the roots of the characteristic equation

The roots of the characteristic equation are complex, that is, in the form

( ± ).

For the general solution,

If the roots of a characteristic equation are in the form  ( ± ), the general solution is given by

From the characteristic equation,

and

Then, the general solution becomes

Now, we will determine

From the question,

y(0) = 1

and

y'(0) = 2

Then,

(NOTE: and )

Then,

∴

Also,

Then,

Recall,

∴

Hence, the solution becomes

ok

step-by-step explanation:

18

step-by-step explanation:

draw the shape and you will see that f and k are actually at adjacent vertices (points of the hexagon that are neighboring).   so find the distance between (1,2) and (4,2).   that is easy since (1,2) and (4,2) is just a vertical distance to count, which is 3.   it is a regular polygon which means all sides are congruent.   you have 6 sides (hexa) so multiply 6 by 3 to receive 18.