Unit-I

Unit-I: First Order Ordinary Differential Equations

Contents

Exact differential equations, Equations reducible to exact form,Linear differential equations, Equations reducible to linear form, Bernoulli's equation.

Differential Equation

Any equation that has a derivative is known as a Differential Equation.

Who discovered Differential Equation?

The development of term 'Differential Equations' came into existence with the invention of calculus by English physicist Isaac Newton and German Mathematician Gottfried Leibniz. Later Swiss Mathematician Jacob Bernoulli proposed Bernoulli Differential Equation. Various other mathematicians like Jean le Rond d'Alembert, Leonhard Euler and Joseph-Louis Lagrange also contributed in discovering various types of differential equations.

Ordinary Differential Equation

An ordinary differential equation is a differential equation containing one or more functions of one independent variable and the derivatives of those functions

Equation Reducible to Exact D.E

Exact Differential Equation

A differential equation of type P(x,y) dx + Q(x,y) dy = 0 is called an exact differential equation, if there exists a function of two variables u(x,y) with continuous partial derivatives such that du = Pdx + Qdy. The general solution of an exact equation is given by u(x,y) = C, where C is arbitrary constant.

Equations Reducible to Exact Form

Integrating Factor:- A function k(x, y) is said to be an Integrating Factor (I. F) of the equation M dx + N dy = 0. If it is possible to obtain a function u(x, y) such that k (M dx + N dy) = du.

I.F is a multiplying factor by which the equation can be made exact.

Linear Differential Equation of the First Order

A differential equation is said to linear if the dependent variable and its derivatives appear only in the first degree.

No term involves power of derivatives or powers of dependent variables or product of derivatives /dependent variables.

OR

Equations Reducible to Linear Form

Bernoulli's Differential Equation

Transformation to Polar

Sometimes it is not possible to solve a differential equation in x, y i.e. by usual method, but can be solved by transforming to polar.

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