Applied Mathematics - Differential Equations

Differential Equations

• Definition, order and degree, general and particular solutions of a differential equation

• Formation of differential equation whose general solution is given

• Solution of differential equations by method of separation of variables solutions of homogeneous differential equations of first order and first degree

• Solutions of linear differential equation of the type −

  • dy/dx + py = q, where p and q are functions of x or constants

  • dx/dy + px = q, where p and q are functions of y or constants

SUMMARY

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

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

3. Degree of a differential equation is defined if it is a polynomial equation in its derivatives.

4. Degree (when defined) of a differential equation is the highest power (positive integer only) of the highest order derivative in it.

5. 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.

6. 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.

7. 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.

8. A differential equation which can be expressed in the form dy/dx f(x, y) or dx/dy g(x, y) where, f (x, y) and g(x, y) are homogenous functions of degree zero is called a homogeneous differential equation.

9. A differential equation of the form dy/dx +py  Q, where P and Q are constants or functions of x only is called a first order linear differential equation.