# How can I find the analytical solution for the system?

• Avan

#### Avan

Member warned about posting with no effort shown

## Homework Statement

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

Omega,N0 and k are positive .

## The Attempt at a Solution

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

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

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

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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.

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.