Formation of Differential Equation and Solutions of a Differential Equation - Practice Questions & MCQ

Formation of Differential Equation

We have studied the general formula of the parabola which is y² = 4ax

This equation represents a family of parabola with a as an arbitrary constant. With different values of a we get a different parabola in this family.

To form its D.E, let us first differentiate it w.r.t. x,

Putting this value of 'a' in original equation of parabola

This is the Differential Equation for family of parabolas y² = 4ax.

Note that there is one arbitrary constant in the original equation, and the order of the D.E. formed is also 1. This is an important result: if an equation contains n arbitary constants, then its D.E. will have the order equal to n.

If we are given a relation between the variables x, y with n arbitrary constants C₁, C₂, .... C_n, then to form its D.E., we differentiate the given relation n times in succession with respect to x, we have altogether n + 1 equations. Now using these we eliminate the n arbitrary constants. The result is a differential equation of the nth order. 

Consider the equation of the family of ellipses with a and b as arbitrary constants

This is the D.E. that represents the given family of ellipses

Note: If arbitrary constants appear in addition, subtraction, multiplication or division, then we can club them to reduce into one new arbitrary constant. Hence, the differential equation corresponding to a family of curves will have order exactly same as number of essential arbitrary constants (number of arbitrary constants in the modified form) in the equation of the curve.

Illustration : 

Here, the number of the arbitrary constant is 3: a, b, and c. But we can club arbitrary constants together  

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Solution of D.E.

If the given D.E. is 

As c is constant of integration, so it can take any real value. We can write it as ln(4a) as well, where a is now the arbitary constant.

This is the general solution of the differential equation, which represents the family of the parabola

If we put some real value of a in this equation, then we get a particular solution of the given differential equation. For example if we put a = 2, then y² = 8x is a particular solution of this equation.

Hence, the solution of the differential equation is a relation between the variables of the equation which satisfy the D.E. and does not contain any derivatives or any arbitary constants.

A general solution of a differential equation is a relation between the variables (not involving the derivatives) which contains the same number of the arbitrary constants as the order of the differential equation. 

Particular solution of the differential equation obtained from the general solution by assigning particular values to the arbitrary constant in the general solution.