Before we start implicit differential equation,first have a look on what is calculus and implicit functions? Calculus is a branch of mathematics that deals with the study of change. Calculus mainly focuses on limits, functions, derivatives, integrals and infinite series. In science, economics, and engineering there are pandemic applications of calculus. Differential calculus and integral calculus are the two branches of calculus.
Differential calculus is the study of the rates at which quantities changes. The primary objects of study in differential calculus are the derivatives of a function, related notions such as the differential and their applications. The derivative of a function describes the rate of change of function near the input value. The producer of finding the derivative is called as differentiation. Now, Implicit functions are those functions in mathematics in which the dependent variable has not been given explicitly in term of the independent variable. If we find the value of x by solving an equation of the form:
R( x, y) = 0.
While in explicit function a prescription is given to determine the value of function y in terms of the input value x:
y = f(x ).
In implicit differentiation, a rule named as chain rule is used to differentiate implicitly defined functions. As I have explained earlier in this page, variable y can be given as a function of x implicitly rather then explicitly. In Implicit differentiation, there is no real distinction between the appearance of x and y. All the implicit equations are be restarted explicitly in one equation.
for example
x2+ y2= 7 is an implicit equation and it needs two explicit equations.
y = + sqrt ( 7 - x2)
and, y = - sqrt(7 - x2).
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