Subset of first-order differential equations

Separable differential equations are a type of first-order differential equations that can be solved analytically

Reformulation to the form \(y' = f(x)g(y)\)

Separable differential equations can be rewritten in the form \(y' = f(x)g(y)\) to separate the variables \(x\) and \(y\)

Integration of each side of the equation independently

The integration of each side of the equation independently leads to the general solution of separable differential equations

Expressing equations in the separable form

To determine if a differential equation is separable, it must be expressed in the form \(y' = f(x)g(y)\)

Algebraic manipulation and logarithmic identities

Some equations may not appear separable at first glance, but can be transformed into the separable form through algebraic manipulation or the use of logarithmic identities

Inherently non-separable equations

Equations with inseparable combinations of variables, such as \(y' = \ln(x + y)\), are inherently non-separable

Modeling dynamic systems

Separable differential equations are used to model dynamic systems in various scientific and engineering fields

Examples of applications

Separable differential equations are applied in scenarios such as mixing problems, pharmacokinetics, environmental changes, economic forecasting, and Newton's Law of Cooling

Importance of identifying constant solutions

Constant solutions, represented by horizontal lines in a direction field, provide insights into the steady states of a system and must be identified when solving separable differential equations

Separation of variables technique

The separation of variables technique involves dividing the differential equation by \(g(y)\) and integrating both sides to obtain an implicit solution

Utilizing \(u\)-substitution

\(u\)-substitution, where \(u = g(y)\) and \(\mathrm{d}u = g'(y) \mathrm{d}y\), is used to evaluate the integrals in the separation of variables technique

Consideration of the interval of existence

The interval of existence for the solution of a separable differential equation must be considered, as it may be limited by the domains of the functions involved