Differential Equations

An equation containing differential coefficients of the dependent variable with respect to independent variable is called a differential equation. Order of differential equation is the order of the highest order derivative involved in the differential equation. Degree of differential equation can only be defined when it is expressed in the form of a polynomial equation in derivatives. It is equal to the power of the highest order derivative in the equation. A function which satisfies the given differential equation is called its solution. The solution which contains as many arbitrary constants as the order of the differential equation is called a general solution of the differential equation. The solution free from arbitrary constants is called particular solution of the differential equation. To form a differential equation from a given function we differentiate the function successively as many times as the number of arbitrary constants in the given function and then eliminate the arbitrary constants. Variable separable method is used to solve such an equation in which variables can be separated completely, i.e., terms containing y should remain with dy and terms containing x should remain with dx. A differential equation of the form (dy/dx) = f(x, y) or (dy/dx) = g(x, y), where f(x, y) and g(x, y) are both homogeneous functions of degree zero is called homogeneous differential equation. To solve the homogenous differential equation we substitute y = vx and (dy/dx) = v + x.(dv/dx). A differential equation of the type (dy/dx) + Py = Q, where P and Q are either constants or functions of x only is called linear differential equation. To solve linear differential equation we first find the Integrating Factor.

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