How can I find the analytical solution for the system?

[Avan]

Homework Statement

dx(t)/dt = N0*sin(omega*t) * x(t) - ( N0*x^2 / k )

Omega,N0 and k are positive .

Homework Equations

The Attempt at a Solution

I tried to solve it using the Bernoulli equations but I could not get the last result.

 

[LCKurtz]

Show us what happens with the Bernoulli substitution so we can see what your difficulty is.

 

[Mark44]

Here's the DE in a bit more readable form.
$$ \frac{dx}{dt} = N_0x(\sin(\omega t) - \frac{x}{k})$$

 

[Avan]

The result that I got is :

exp( (cos(omega*t) * (-N0/omega)) / x(t) = (N0 / k) * ( integral (exp( (cos(omega*t)) * (-N0/omega) ) ) ) dt

So I do not know how to find the integral when typically there is no specific solution for the integral of the Exponential function.

 

[LCKurtz]

You may just have to express the result in terms of unevaluated integrals. Integrals of the form $\int e^{\frac 1 \omega \cos(\omega t)}~dt$ don't have elementary antiderivatives.