A differential equation is separable if it can be algebraically manipulated so that all terms involving the dependent variable (typically y) and its differential (dy) are on one side of the equation, and all terms involving the independent variable (typically x) and its differential (dx) are on the other side.
In other words, a first-order differential equation of the form dy/dx = f(x, y)
is separable if f(x, y)
can be expressed as a product of two functions, one depending only on x and the other depending only on y. This can be written as:
dy/dx = g(x)h(y)
Where:
g(x)
is a function of x only.h(y)
is a function of y only.
How Separability Works:
-
Identify the Differential Equation: Start with a differential equation in the form
dy/dx = f(x, y)
. -
Check for Separability: Determine if
f(x, y)
can be factored intog(x)h(y)
. -
Separate the Variables: Algebraically rearrange the equation so that all y terms are on one side (with dy) and all x terms are on the other side (with dx). This will result in an equation like:
dy/h(y) = g(x) dx
orf(y) dy = g(x) dx
-
Integrate Both Sides: Integrate both sides of the separated equation with respect to their respective variables.
∫f(y) dy = ∫g(x) dx
-
Solve for y: Solve the resulting equation for y to obtain the general solution of the differential equation. This may involve algebraic manipulation.
Examples:
Example 1: Separable
Consider the differential equation: dy/dx = x/y
- We can rewrite this as:
y dy = x dx
- Integrating both sides:
∫y dy = ∫x dx
- This gives:
y²/2 = x²/2 + C
- Solving for y:
y = ±√(x² + 2C)
Example 2: Not Separable
Consider the differential equation: dy/dx = x + y
- We cannot separate x and y into a product of
g(x)
andh(y)
. The addition operation prevents separation. Therefore, this differential equation is not separable.
Why Separability Matters:
Separable differential equations are important because they are a relatively simple type of differential equation that can be solved directly by integration. They are a fundamental building block in the study of differential equations and serve as an introduction to more complex solution techniques.