We are here going to begin looking at nonlinear first order differential equations. The first type of nonlinear first order differential equations which we will see is separable differential equations.
A separable differential equation is any differential equation which we can write in the subsequent form.
N(y)(dy/dt) = M (x)
Remember that in order for a differential equation to be separable all the y's in the differential equation should be multiplied through the derivative and all the x's in the differential equation should be on the other side of the equivalent sign.
Resolving separable differential equation is quite easy. We initially rewrite the differential equation as the subsequent:
N ( y ) dy = M ( x ) dx
After that you integrate both sides.
∫ N ( y ) dy = ∫M ( x ) dx
Therefore, after doing the integrations in (2) you will contain an implicit solution which you can hopefully resolve for the explicit solution, y(x). Remember that this won't always be possible to resolve for an explicit solution.
Recall from the Definitions section which an implicit solution is a solution which is not in the form y = y ( x) whereas an explicit solution has been written in that form.
We will also have to worry regarding the interval of validity for several of these solutions. Recall such the interval of validity was the range of the independent variable, x in such case, on that the solution is valid. Conversely, we need to ignore division via zero, complicate numbers and logarithms of negative numbers or zero etc. Most of the solutions which we will find from separable differential equations will not be valid for each value of x.