How can I prove this function is a periodic phenomenom?

I´ve been stuck with this math problem for a while and I would really appreciate it if someone could help me with this problem. Any tips or suggestions is fine, I would be really thankful.

So I´ve this differential equation here: 

Then I know that if the discriminant of the characteristic equation is negative, it´ll give me two complex solutions and it´ll describe a periodic phenomenon.

I´ve to prove that it´s the case with this equation: 

But I really don´t know how to start this? I´ve a general idea that if I get a two complex roots then I can find the modulus of the complex number, theta and also a point cos (x) and sin (y) on the plane. But how do I really prove this to be the case mathematically?

Expert Answers

An illustration of the letter 'A' in a speech bubbles

The periodic phenomena are very well modelled by the second order differential equations. In the provided case, you need to  start solving the characteristic equation for the given differential equation, such that:

`r^2 - alpha(beta+1)r + alpha*beta = 0`

Using quadratic formula, yields:

`r_(1,2) = (alpha(beta+1) +- sqrt(alpha^2(beta+1)^2 - 4alpha*beta))/2`

The quadratic equation has two different real solutions if the radicand `alpha^2(beta+1)^2 - 4alpha*beta > 0` , two equal real solutions if `alpha^2(beta+1)^2 - 4alpha*beta = 0` and complex conjugate solutions if `alpha^2(beta+1)^2 - 4alpha*beta < 0` .

You need to get a general solution to the given differential equation, such that:

`y = y_c + y_p`

`y_c` represents the complementary solution

`y_p` represents the particular solution

The complementary solution can be evaluated solving the characteristic equation `r^2 - alpha(beta+1)r + alpha*beta = 0` .

If the solutions `r_(1,2)` are different and real, then the complementary solution is the following:

`y_c = c_1*e^(r_1*t) + c_2*e^(r_2*t)`

If the solutions `r_(1,2) = a+-b*i` are complex, then the complementary solution is the following:

`y_c = c_1*e^(a*t)cos(b*t) + c_2*e^(a*t)sin(b*t) `

The problem does not provide the equation of the function `G` , but you may solve the problem to evaluate the constants `c_1` and `c_2` using the method of undetermined coefficients.

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial