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An equation involving derivatives of the dependent variable with respect to independent variable is known as a differential equation.

Order of Differential Equation

Order of a differential equation is the order of the highest order derivative occurring in the differential equation.

Degree of Differential Equation

Degree of a differential equation is defined if it is a polynomial equation in its derivatives. Degree of a differential equation is the highest power (positive integer only) of the highest order derivative in it.

Solution of Differential 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 and the solution free from arbitrary constants is called particular solution.

Formation of 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

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 first order-first degree differential equation is of the form

dy/dx = F(x,y)

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

dy/dx = h(y) . g(x)

1/h(y) dy = g(x) dx

∫ 1/h(y) dy = ∫ g(x) dx

H(y) = G(x) + C

### Linear Differential Equations

Linear differential equation is of the from

dy/dx + Py = Q where, P and Q are constants or functions of x only.

To solve the first order linear differential equation, multiply both sides of the equation by a function of x

g(x) dy/dx + P . (g(x)) y =Q . g(x)

g(x) = ∫P dx

Integrating Factor (I.F.)= e ∫P dx

The solution of the given differential equation

y (I.F.) = ∫ Q (I.F.) dx + c