Differential Equations

First Order Equations

1st Order Diff Equations Logo

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 as the ratio of two differentials we move to the right side and divide the equation by

Of course, we need to make sure that If there's a number such that then this number will also be a solution of the differential equation. Division by causes loss of this solution.

By denoting we write the equation in the form

We have separated the variables so now we can integrate this equation:

where is an integration constant.

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 and We divide the equation by and move to the right side:

One can notice that after dividing we can lose the solutions and when becomes zero. In fact, let's see that is a solution of the differential equation. Obviously,

Substituting this into the equation gives Hence, is one of the solutions. Similarly, we can check that is also a solution.

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: Thus, the final solution of the equation is written in the form

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 where is a constant. However, it is possible to do not for all differential equations.

Example 2.

Solve the differential equation

Solution.

We can rewrite this equation in the following way:

Divide both sides by to get

Obviously, that for all real Check if is a solution of the equation. Substituting and into the differential equation, we see that the function is one of the solutions of the equation.

Now we can integrate it:

Notice that Hence,

We can represent the constant as where Then

Thus, the given differential equation has the following solutions:

This answer can be simplified. Indeed, if using an arbitrary constant which takes values from to the solution can be written in the form:

When , it becomes

See more problems on Page 2.

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