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

dy/dx=  h(y).g(x)

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

log(tany)= -log(tanx)+logC

tany.tanx=C

This is required general solution of differential equation.

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