Separable Equations
A first order differential equation y' = f (x, y) is called a separable equation if the function f (x, y) can be factored into the product of two functions of x and y:
where p (x) and h (y) are continuous functions.
Considering the derivative
Of course, we need to make sure that
By denoting
We have separated the variables so now we can integrate this equation:
where
Calculating the integrals, we get the expression
representing the general solution of the separable differential equation.
Solved Problems
Example 1.
Solve the differential equation
Solution.
In the given case
One can notice that after dividing we can lose the solutions
Substituting this into the equation gives
Returning to the differential equation, we integrate it:
We can calculate the left integral using the fractional decomposition of the integrand:
Thus, we get the following decomposition of the rational integrand:
Hence,
We can rename the constant:
Here the general solution is expressed in implicit form. In the given case we can transform the expression to obtain the answer as an explicit function
Example 2.
Solve the differential equation
Solution.
We can rewrite this equation in the following way:
Divide both sides by
Obviously, that
Now we can integrate it:
Notice that
We can represent the constant
Thus, the given differential equation has the following solutions:
This answer can be simplified. Indeed, if using an arbitrary constant
When