Variable Separation Method

We learnt about differential equations in this . Today we will take a look at a very basic method of solving differential equations by separating the variables.

If a differential equation can be rearranged to the form $f(x)dx = g(y) dy$ then the equation is called a variable separable equation.

These are generally solvable using direct integrations of the form:

$\displaystyle \int f(x)dx = \int g(y)dy + C$

Let us take the example of the following differential equation:

$(1 + x) y dx = (1 + y) x dy$

The above equation can be rearranged as:

$\dfrac{1 + x}{x}dx = \dfrac{1 + y}{y}dy$

Integrating both sides:

$\displaystyle \int \left(\dfrac{1 + x}{x}\right)dx = \int \left(\dfrac{1 + y}{y}\right) dy$

$\Rightarrow \displaystyle \int \left(\dfrac{1}{x} + 1\right)dx = \int \left(\dfrac{1}{y} + 1\right) dy$

$\Rightarrow \displaystyle \int \dfrac{dx}{x} + \int dx = \int \dfrac{dy}{y} + \int dy$

$\Rightarrow \displaystyle \ln x + x = \ln y + y + C$

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$\Rightarrow \displaystyle \ln x - \ln y = y - x + C$

$\Rightarrow \ln \left( \dfrac{x}{y} \right) = y - x + C$

$\Rightarrow \dfrac{x}{y} = e^{y - x + C}$

$\Rightarrow \dfrac{y}{x} = e^{x - y - C}$

$\Rightarrow \dfrac{y}{x} = e^{x - y} \cdot e^{-C}$

$\because C$ is a constant, substituting $e^{-C} = A$, in the above equation, we get:

$\dfrac{y}{x} = A e^{x - y}$

$\Rightarrow y = Axe^{x - y}$