The method of separation of variables is a technique for solving differential equations where the variables are separated and treated independently.
Consider a differential equation in the form:
[f(x) dx = g(y) dy]
To separate the variables, we move all the (x) terms to the left side of the equation and all the (y) terms to the right side:
[f(x) dx = g(y) dy]
Now, we divide both sides of the equation by the respective function to isolate the variables:
[\frac{1}{g(y)} dy = \frac{1}{f(x)} dx]
This allows us to integrate the variables separately. Integrating both sides, we have:
[\int \frac{1}{g(y)} dy = \int \frac{1}{f(x)} dx]
After integration, we obtain:
[G(y) = F(x) + C]
Here, (G(y)) and (F(x)) represent the antiderivatives (or indefinite integrals) of (g(y)) and (f(x)), respectively, and (C) is an integration constant.
The solution to the original differential equation then involves substituting the expressions for (G(y)) and (F(x)) back in and solving for the desired variable.
Certainly! Here’s a concrete example to illustrate the method of separation of variables:
Let’s consider the differential equation:
[\frac{dy}{dx} = x^2 \cdot y]
To solve this equation using separation of variables, we can rearrange it as:
[\frac{1}{y} \cdot dy = x^2 \cdot dx]
Now we separate the variables by moving the (y) terms to the left side and the (x) terms to the right side:
[\frac{1}{y} \cdot dy = x^2 \cdot dx]
Next, we integrate both sides of the equation with respect to their respective variables. Integrating the left side gives:
[\int \frac{1}{y} \cdot dy = \ln | y | ] |
And integrating the right side gives:
[\int x^2 \cdot dx = \frac{x^3}{3} + C]
Combining the results, we have:
[\ln | y | = \frac{x^3}{3} + C] |
where (C) is the constant of integration.
To solve for (y), we can exponentiate both sides of the equation:
[ | y | = e^{\frac{x^3}{3} + C}] |
Since (e^C) is just a positive constant, we can rewrite the equation as:
[y = \pm e^{\frac{x^3}{3}} \cdot k]
where (k) is a non-zero constant.
So, the general solution to the given differential equation is:
[y = \pm e^{\frac{x^3}{3}} \cdot k]
where (k) can take any non-zero value.