|Class 12 Maths Differential Equations||Differential Equations with Variables Separable|
Differential Equations with Variables Separable
A first order-first degree differential equation is of the form
dy/dx = F(x,y) --(1)
If F (x, y) can be expressed as a product g (x) h(y), where, g(x) is a function of x and h(y) is a function of y, then the differential equation (1) is said to be of “variable separable” type. Then differential equation (1) has the form
d(y)/h(y) = g(x). dx ... (3) where h(y) ≠ 0
Integrating both sides of equation (3), we get
∫ (1/h(y)).dy = ∫ g(x). dx = ... (4)
Thus, (4) provides the solutions of given differential equation in the form
H(y) = G(x) + C
Here, H (y) and G (x) are the anti-derivatives of 1/h(y) and g(x) respectively and C is the arbitrary constant.
Problem: find the general solution of the following of question : sec2 x tan y dx + sec2 y tan x dy = 0
Solution: Sec2y tanx dy= - sec2 x tan y dx
Integrating both side
Let tany=t and sec2ydy = dt
This is required general solution of differential equation.