Introduction to Differential Equations

Definition : A differential equation is an equation containing an unknown function and its derivatives. Examples:. y is dependent variable and x is independent variable, and these are ordinary differential equations Ordinary Differential Equations

Order of Differential Equation The order of the differential equation is order of the highest derivative in the differential equation. Differential Equation ORDER 1 2 3

Degree of Differential Equation Differential Equation Degree The degree of a differential equation is power of the highest order derivative term in the differential equation.

Solution of Differential Equation

Solving a differential equation means finding an equation with no derivatives that satisfies the given differential equation. Solving a differential equation always involves one or more integration steps. It is important to be able to identify the type of differential equation we are dealing with before we attempt to solve it.

Direct Integration

Separation of Variables Some differential equations can be solved by the method of separation of variables (or "variables separable"). This method is only possible if we can write the differential equation in the form A(x) dx + B(y) dy = 0, where A(x) is a function of x only and B(y) is a function of y only. Once we can write it in the above form, all we do is integrate throughout, to obtain our general solution.

Differential Equation Chapter 1

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Differential Equation Chapter 1 18 y = A. ln (x)

Particular Solutions Our examples so far in this section have involved some constant of integration, K. We now move on to see particular solutions, where we know some boundary conditions and we substitute those into our general solution to give a particular solution.

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Homogeneous Equations Sometimes, the best way of solving a DE is a to use a change of variables that will put the DE into a form whose solution method we know. We now consider a class of DEs that are not directly solvable by separation of variables, but, through a change of variables, can be solved by that method.

If a function M(x, y) has the property that : M(tx, ty) = t n M(x, y), then we say the function M(x, y) is a ho- mogeneous function of degree n

A differential equation M(x, y)dx + N(x, y)dy = 0 is said to be a homogeneous differential equation if both functions M(x, y) and N(x, y) are homogeneous functions of (the same) degree n. In this case, we can represent the functions as M(x, y) = x n M(1, y/x) N(x, y) = x n N(1, y/x)

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