Differential equation

We want to solve the following differential equation \(E\):

\[y'(t) = STP(t) - \alpha\cdot exp(-\lambda \cdot t) + (\omega_P-\omega_M) \cdot y(t)\]

with:

\[STP(t) = \delta\left(\frac{t}{\tau}\cdot exp(1-\frac{t}{\tau})\right)^2\]

The general solution of differential equation \(E_0\):

\[y'(t) - (\omega_P-\omega_M) \cdot y(t) = 0\]

is \(y_G = Kexp((\omega_P-\omega_M) t)\)

A particular solution of differential equation \(E_1\):

\[y'(t) - (\omega_P-\omega_M) \cdot y(t) = \alpha\cdot exp(-\lambda\cdot t)\]

is

\[y_{p1} = -\frac{\alpha}{(\omega_P-\omega_M)+\lambda} \cdot exp(-\lambda\cdot t)\]

A particular solution of differential equation \(E_2\):

\[y'(t) - (\omega_P-\omega_M) \cdot y(t) = \delta \left(\frac{t}{\tau}exp(1-\frac{t}{\tau})\right)^2\]

is

\[y_{p2} = exp(1-\frac{t}{\tau})^2\frac{2\delta}{\tau^2(-2(\omega_P-\omega_M)-\tau)}\left(t^2 - t\frac{4}{(-2(\omega_P-\omega_M)-\tau)} + \frac{8}{(-2(\omega_P-\omega_M)-\tau)^2}\right)\]

The general solution to differential equation (E) is therefore:

\[y(t) = Kexp(-(\omega_P-\omega_M) t) -\frac{\alpha}{(\omega_P-\omega_M)+\lambda} \cdot exp(-\lambda\cdot t) + exp(1-\frac{t}{\tau})^2\frac{2\delta}{\tau^2(-2(\omega_P-\omega_M)-\tau)}\left(t^2 - t\frac{4}{(-2(\omega_P-\omega_M)-\tau)} + \frac{8}{(-2(\omega_P-\omega_M)-\tau)^2}\right)\]

\[ K = y_0+ \frac{\alpha}{(\omega_P-\omega_M)+\lambda} -\frac{16e^2\delta}{\tau^2(2(\omega_P-\omega_M)-\tau)^3}\]