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

if x = 1, then the answer is 3.5