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