Monday, August 13

Types of Differential Equations



Differential equation is an equation which involves derivatives or differential coefficient or differentials. In mathematics differential equation shows the relationships between physical quantities which are appear in mathematical form.

Differential equations are ordinary and partial type. In ordinary differential equation all the derivatives coefficient with respect to single variable and in partial differential equation two or more independent variables with respect to any of them. Homogenous differential equations are in the form of dY/dX=F(X, Y). homogenous function means all powers are same. F(X, Y) should be homogenous function of the same degree in x and y. suppose given an expression like (aX^n+ bX^n-1+ cX^n-2+ --------so on) in which every term is of the nth degree, is called a homogenous function of degree n. this can be rewritten as by taking (X^n) common. Thus any function F(X, Y) which can be expressed in the form X^nF(X/Y) is called a homogenous function of degree n in X and Y. for example X^3cos(X/Y) is a homogeneous function of degree 3 in X and Y.

For solving homogeneous differential equations, there are some different methods. To solve homogeneous differential equation we use following steps
1 we put Y=VX, then differentiate Y with respect to X. by differentiating we get dY/dX=V+dY/dX.
2 now we separate the variables X and V and then integrate.

If homogeneous differential equation with constant coefficient then, we have to find its complementary function and particular integral. Then we get complete solution which is addition of both complementary function and particular integral. Such type of equations is most important in the study of electromechanical vibrations and other engineering problems.

Non homogeneous differential equation means in given function the degree of X and Y are not same. Non homogeneous function, this type of function is product of two different function. Suppose function is F(Y)=X, where X is an exponential function, trigonometric function, high power of X, product of exponential and higher order of X functions, product of exponential and trigonometric function, and product of higher order of X and trigonometric functions.

For solving non homogeneous differential equations, we use complementary function and particular integral method.  Complementary function means we have to write the auxiliary equation of given expression and find its roots. Then we substitute in given equation to find particular integral. Finally we get the complete solution by adding particular integral and complementary function. We can also use variation of parameters, couchy-euler methods.

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