Differential Equations
Definition:
An equation involving one dependent variable and one or more independent variables and the derivatives of dependent variables with respect to independent variable are called a differential equation.
The differential equation which contains one independent variable and ordinary derivatives with respect to this independent variable is said to be an ordinary differential equation.
A differential equation is said to be a partial differential equation if it contains at least two independent variables and partial derivatives with respect to either of these independent variables.
Importance:
The differential equations are used in Physics, Chemistry, Biology, Applied mathematics and many other fields. Especially differential equations are used in many physical laws involved in the rate of the physical quantity. Differential equations play an important role in modeling every physical, technical or biological process. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, but we can calculate the approximate solution.
Differential equations can be divided into different types:
- Ordinary Differential equations
d2y/dx2 + 3(dy/dx) = sinx
- Partial Differential equations
δ2y/δx2 + δy/δx + 4y = 2
Difficulties and techniques to solve the Differential equation:
As there are different methods to solve the given differential equation, it is important to select the suitable method. We may use more than one method to solve the single differential equation. The Student who lacks the concepts of differentiation and integration will have to face the difficulty while solving the question. So the Student needs the concepts of not only the derivatives but also the integration methods though the name indicates solving the differential equation.
We can solve the given differential equation by using different methods:
Variables separable method: By separating the derivatives of the different variables and integrate them separately solve the equation.
Homogeneous method: For the equation dy/dx = f(x,y) / g(x, y) , if the functions are of same degree in x and y then the equation is said to be homogeneous differential equation. By substituting one variable in terms of other variable and then apply variable separable method to solve the equation.
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