1. Idea of Variables Separable Method
The variables separable method is one of the simplest ways to solve first-order differential equations. The idea is to rearrange the equation so that all terms involving \(y\) appear on one side and all terms involving \(x\) appear on the other side. Once this separation is done, the equation can be integrated directly.
The method works for many equations that can be expressed in the form where the derivative can be written as a product of one function of \(x\) and another function of \(y\).
1.1. When separation is possible
Separation works when the differential equation can be rewritten as:
\(f(y)\, dy = g(x)\, dx.\)
Not every differential equation can be separated, but many practical ones can.
2. Definition of a Separable Differential Equation
A differential equation is called separable if it can be expressed in the form:
\(\frac{dy}{dx} = F(x)G(y),\)
or can be rearranged to that form by algebraic manipulation. This allows moving all terms with \(y\) to the left and all terms with \(x\) to the right.
2.1. Rewriting into separable form
If the equation is written as
\(\frac{dy}{dx} = \frac{g(x)}{f(y)},\)
then multiplying both sides by \(f(y)dx\) gives
\(f(y)\, dy = g(x)\, dx,\)
which is already in separable form.
3. Steps for Solving a Separable Equation
To solve a separable differential equation, follow these steps:
3.1. Step-by-step solution process
- Rewrite the equation so that terms involving \(y\) appear on one side and terms involving \(x\) appear on the other.
- Integrate both sides separately with respect to their variables.
- Add the constant of integration after integrating.
- If possible, simplify the result and write \(y\) explicitly.
4. General Solution Form
Once separation is achieved, the general solution takes the form:
\(\int f(y)\, dy = \int g(x)\, dx + C,\)
where \(C\) is an arbitrary constant. Sometimes the solution can be solved explicitly for \(y\); in other cases, leaving the implicit form is acceptable.
5. Worked Examples
These examples illustrate how the variables separable method works in practice.
5.1. Example 1: Simple separable equation
Solve:
\(\frac{dy}{dx} = x y.\)
Separate the variables:
\(\frac{1}{y} dy = x dx.\)
Integrate both sides:
\(\int \frac{1}{y} dy = \int x dx.\)
This gives:
\(\ln|y| = \frac{x^2}{2} + C.\)
Exponentiate:
\(y = A e^{x^2/2}.\)
5.2. Example 2: Rational form
Solve:
\(\frac{dy}{dx} = \frac{x^2}{1 + y^2}.\)
Separate the variables:
\((1 + y^2) dy = x^2 dx.\)
Integrate:
\(\int (1 + y^2) dy = \int x^2 dx.\)
Results:
\(y + \frac{y^3}{3} = \frac{x^3}{3} + C.\)
This implicit equation represents the general solution.
5.3. Example 3: Involving roots
Solve:
\(\frac{dy}{dx} = \sqrt{1 - y^2}.\)
Separate variables:
\(\frac{dy}{\sqrt{1 - y^2}} = dx.\)
Integrate both sides:
\(\sin^{-1}(y) = x + C.\)
Thus:
\(y = \sin(x + C).\)
6. Notes and Observations
Important things to keep in mind:
- Not every differential equation is separable, but many common ones are.
- Check if algebraic manipulation can make the equation separable.
- Implicit solutions are acceptable if explicit isolation of \(y\) is difficult.
- The constant of integration must always be included after integrating.