This document contains mathematical formulas to remember.

All differential equations of the first order and first degree cannot be solved. Only those among them which belong to (or can be reduced to) one of the following categories can be solved by the standard methods. 

In this chapter, we shall discuss such physical problems which lead to differential equations of the first order and first degree. A summary of the fundamental principles required in the formation of such differential equations is given in each case.

A linear differential equation is that in which the dependent variable and its derivatives occur only in the first degree and are not multiplied together. 

In this chapter, we shall study those physical problems which deal with the linear differential equations of higher order with constant coefficients. Such equations play a dominant role in unifying the theory of electrical and mechanical oscillatory system, simple harmonic motion, deflection of beam, simple pendulum and population models. We shall begin by explaining the phenomenon of simple harmonic motion and then electric circuits.
 

A transformation is a mathematical device which converts one function into another. For example, when the differential operator D ? F H G I K J d dx operates on f(x) = sin x, it gives a new function g(x) = D f(x) = cos x. Laplace transform or Laplace transformation is widely used by scientists and engineers. It is particularly effective in solving linear differential equations—ordinary as well as partial. It reduces an ordinary differential equation into an algebraic equation. 

Laplace transforms can be used to solve ordinary as well as partial differential equations. We shall apply this method to solve only ordinary linear differential equations with constant co-efficients. The advantage of this method is that it yields the particular solution directly without the necessity of first finding the general solution and then evaluating the arbitrary constants. 

If a quatity z has a unique, finite value for every pair of values of x and y, then z is called a functoion of two variables x and y.

A function f(x, y) is said to have a maximum value at x = a, y = b if f(a, b) > f(a + h, b + k), for small and independent values of h and k, positive or negative. A  function  f(x, y)  is  said  to  have a minimum value at x = a, y = b if f(a, b) < f(a + h, b + k), for small and independent values of h and k, positive or negative. Thus f(x, y) has a maximum or minimum value at a point (a, b) according as ?f = f(a + h, b + k) – f(a, b) < or > 0. 

A differential equation which involves partial derivatives is called a partial differential equation.

The order of a partial differential equation is the order of the highest ordered partial derivative in the equation. The degree of a partial differential equation is the degree of the highest order partial derivative occurring in the equation. Thus, equation (1) is of first order, equations (2) and (3) are of second order. The degree of all the above equations is one. 

The complementary function is the general solution of ?(D, D?) = 0. It must contain n arbitrary constants, where n is the order of the differential equation. The particular integral is a particular solution (free from arbitrary constants) of ?(D, D?)z = F(x, y). The complete solution of (1) is z = C.F. + P.I.
 

This document contains content table for the book A Textbook for Engineering Mathematics Semester I.

This document contains preface to A Textbook For Engineering Mathematics Semester I.

This document contains syllabus of Engineering Mathematics Semester I.