Sunday, April 27, 2008

Differential Equations - Part 1

Differential equation is an equation involving derivatives of a dependent variable with respect to one or more independent variables.

Example: d²y/dx²+ y = x²

Order and degree are two attributes of a differential equation.

The order of a differential equation is the order of the highest differential coefficient involved. If second order derivative is present in the differential equation, the order of the equation is two or it is of second order. The equation give above as an example is a second order equation as d²y/dx² a second order derivative of y is present in the equation.

In the equation, the power to which the higher differential coefficient or derivative is raised is known as the degree of the equation.

Examples (d³y/dx³)4 + (d²y/dx²)² +y² = 0 is a 3rd order and 4th degree equation. 3rd order because d³y/dx³ is present in the equation and it is the highest order derivative in the equation. 4th degree because d³y/dx³ is raised to the fourth power.

Formation of a differential equation

If a polynomial f(x,y,c1, is the solution of a differential equation, where x and y are variables and c1, are constants, differentiate the equation n times successively and eliminate the constants.

Solution of a differential equation
It is f(x,y, c1, c2, which does not involve derivatives and the derivatives of f(x,y,c1, satisify the differential equation.

Solution of first order and first degree differential equations

Equations in which the variables are separable

If the given differential equation can be written in the form

f(x)dx + g(y)dy = 0

we can see that variables x and y are separated in the left hand side expression.

We can write the equation as

f(x)dx = -g(y)dy

integrating both sides we get the solution.

Soluton of linear differential equation

In a linear differential equation, the dependent variable and its derivatives occur in the first power only and are not multiplied together.

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