# Differential Equations – Variables Separable Method

## Differential Equations – Variables Separable Method

Equations in which the variables are separable are those equations which can be expressed that the coefficient of dx is only a function of x and that of dy is only a function of y.

Thus, the general form of such an equation is f(x) dx + g(y) dy = 0. The solution of this equation is obtained by integrating f(x) and g(y) with respect to x and y respectively. i.e., solution is given by

∫f(x) dx = ∫ g(y) dy + C

Example: Find the solution of the differential equation sec²x. tan y dx + sec²y.tanx dy = 0

Solution: Given, sec²x tany dx + sec²y tanx dy = 0

On separating the variables, we get

sec²x tany dx = -sec²y tanx dy

$$\frac{{{\sec }^{2}}x}{\tan x}dx=-\frac{{{\sec }^{2}}y}{\tan y}dy$$,

On integrating both sides, we get

$$\int{\frac{{{\sec }^{2}}x}{\tan x}dx}=-\int{\frac{{{\sec }^{2}}y}{\tan y}dy}$$,

Let us consider

tan y = v

$${{\sec }^{2}}y=\frac{dv}{dy}$$,

$$dy=\frac{dv}{{{\sec }^{2}}y}$$,

$$\int{\frac{{{\sec }^{2}}x}{u}\frac{du}{{{\sec }^{2}}x}}=-\int{\frac{{{\sec }^{2}}y}{v}}\frac{dv}{{{\sec }^{2}}y}$$,

$$\int{\frac{du}{u}=-\int{\frac{dv}{v}}}$$,

log |u| = -log |v| + log |C|

log |tanx| = -log |tany| + log |C|

log |tanx tany| = log |C| (∵ log m + log n = logmn)

tanx. tany = C (∵ logm = logn → m = n)

Which is the required general solution.