Solve y ' ' + 4 y = 0 , y ( 0 ) = 2 , y ' ( 0 ) = 2 The resulting oscillation will have Amplitude: Period: If your solution is A sin(rt)+Bcos(rt). The amplitide is given by √ A 2 + B 2 and the period is given by 2 π r

Step-by-step explanation:

This is a homogeneous linear equation. So, assume a solution will be proportional to:

Now, substitute into the differential equation:

Using the characteristic equation:

Factor out

Where:

Therefore the zeros must come from the polynomial:

Solving for :

These roots give the next solutions:

Where and are arbitrary constants. Now, the general solution is the sum of the previous solutions:

Using Euler's identity:

Redefine:

Since these are arbitrary constants

Now, let's find its derivative in order to find and

Evaluating     :

Evaluating     :

Finally, the solution is given by:

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step-by-step explanation: