Differential Equations
Basic Concepts of Differential Equations
* Differential equation: an equation involving a derivative of the dependent variable with respect to the independent variable is called a differential.
* The order of the highest order derivative occurring in a differential equation is called the order of the differential equation.
* The degree of a polynomial differential equation is the degree of its highest derivatives.
* The order and the degree of a differential equation are always positive integers.
Solution of Differential Equation
* The solution of a differential equation is the function that satisfies it.
* The two types of solution for a differential equation:
* General Solution
* Particular Solution
Forming a Differential Equation
* Working method to obtain a differential equation from a given general solution:
1. Observe the number or arbitrary constants present in the given general solution.
2. Find the derivative up to the order based on the number of constants.
3. Eliminate the arbitrary constants using the equation obtained.
Differential Equation
* Differential Equation can be used to represent a family of curves.
* Method to obtain a differential equation for a family of curves:
1. Select the general equation of the required curve with unknown number of the parameters F1(x, y, a) = 0 or F2 (x, y, a, b) = 0.
2. Differentiate the general equation with respect to x. This equation has to be differentiated again based on the number of parameters.
3. Eliminated the unknown parameters to obtain the differential equation.
Variables Separable Method
The variables separable method is one of the methods to solve first order first degree differential equations. This method is applicable when the variables can be
separated.
Homogeneous Differential Equation
* If F (x, y) is a homogeneous function of degree n, then
F (kx, ky) = kn F (x, y), where k is a constant.
* Representation of a homogeneous function:
F (x, y) = x^{n} g \begin{pmatrix}
y \\
x
\end{pmatrix}
F (x, y) = y^{n} g \begin{pmatrix}
x \\
y
\end{pmatrix}
n = Degree of homogeneous Equation
* A differential equation of the form \frac{dy}{dx} = F (x, y) is said to be homogeneous
If F (x, y) is a homogeneous function of degree zero
* A homogeneous differential equation is solved by substituting
y = vx or x = vy.
Linear Differential Equations
* The general solution of a linear differential equation \frac{dy}{dx} + Py = Q is given by y ×
(Integrating Factor) = ∫ Q x (Integrating Factor) dx,
* Here, Integrating Factor = e ∫ P dx
* The general solution of a linear differential equation \frac{dx}{dy} + Px = Q is given by x ×
(Integrating Factor) = ∫ Q x (Integrating Factor) dy,
Here, Integrating Factor = e ∫ P dy