Engineering Mathematics II - Formation of Differential Equations

Ordinary and Partial Differential Equations

• A differential equation involves one dependent variable and its derivatives with respect to one or more independent variables.
• An ordinary differential equation (ODE) contains only one independent variable and hence ordinary derivatives.
• A partial differential equation contains two or more independent variables and hence partial derivatives.

Order and Degree of a Differential Equation

• The order of a differential equation is the order of the highest derivative in it.
• The degree of a differential equation is the largest exponent of the highest order derivative.
• If the differential equation cannot be expressed as a polynomial of derivatives, its degree is not defined.

Formation of Ordinary Differential Equations

• To form an ODE, differentiate the given equation with respect to the independent variable, eliminate the arbitrary constant, and get the required differential equation.
• If the equation involves multiple arbitrary constants, differentiate the equation multiple times, eliminate the constants, and get the required differential equation.

Examples of Formation of Ordinary Differential Equations

• If the general solution is of the form y = e2x + x + C, the differential equation formed is y' = 2e2x + 1.
• If the general solution is of the form F(x, y, a) = 0, the differential equation formed is obtained by differentiating the equation once with respect to the independent variable, eliminating the arbitrary constant, and getting the required differential equation.
• If the general solution is of the form F(x, y, a, b) = 0, the differential equation formed is obtained by differentiating the equation twice with respect to the independent variable, eliminating the arbitrary constants, and getting the required differential equation.

Examples of Differential Equations

• The differential equation of the family of concentric circles is x + y dy/dx = 0.
• The differential equation of the family of circles x2 + y2 = 2ax is 2xy dy/dx = y2 - x2.
• The differential equation of the family of curves y = asin(x+b) is d2y/dx2 + y = 0.
• The differential equation of the family of curves y = a/x + b is xy' - y = 0.
• The differential equation of the family of curves y = ae4x + be−x is y''' - 4y'' + 7y' - 4y = 0.

This quiz covers the basics of differential equations, including ordinary and partial differential equations, and their formation in Engineering Mathematics II.