Examples of Differential Equations - Separable (homogeneous) First-order Linear Ordinary Differential Equations

Separable (homogeneous) First-order Linear Ordinary Differential Equations

A separable linear ordinary differential equation of the first order must be homogeneous and has the general form

where is some known function. We may solve this by separation of variables (moving the y terms to one side and the t terms to the other side),

Since the separation of variables in this case involves dividing by y, we must check if the constant function y=0 is a solution of the original equation. Trivially, if y=0 then y'=0, so y=0 is actually a solution of the original equation. We note that y=0 is not allowed in the transformed equation.

We solve the transformed equation with the variables already separated by Integrating,

where C is an arbitrary constant. Then, by exponentiation, we obtain

.

Here, so . But we have independently checked that y=0 is also a solution of the original equation, thus

.

with an arbitrary constant A, which covers all the cases. It is easy to confirm that this is a solution by plugging it into the original differential equation:

Some elaboration is needed because ƒ(t) might not even be integrable. One must also assume something about the domains of the functions involved before the equation is fully defined. The solution above assumes the real case.

If is a constant, the solution is particularly simple, and describes, e.g., if, the exponential decay of radioactive material at the macroscopic level. If the value of is not known a priori, it can be determined from two measurements of the solution. For example,

gives and .