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.