# Formation of Differential Equation

Here you will learn formation of differential equation with examples.

Let’s begin –

## Formation of Differential Equation

Algorithm

1). Write the equation involving independent variable x (say), dependent variable y (say) and the arbitrary constants.

2). Obtain the number of arbitrary constants in Step 1. Let there be n arbitrary constants.

3). Differentiate the relation in step 1 n times with respect to x.

4). Eliminate arbitrary constants with the help of n equations involving differential coefficients obtained in step 3 and an equation in step 1. The equation so obtained is the desired differential equation.

Example : Form the differential equation of the family of curves represented by $$c(y + c)^2$$ = $$x^3$$ , where c is a parameter.

Solution : The equation of the family of curves is $$c(y + c)^2$$ = $$x^3$$                            ……….(i)

Clearly, it is one parameter family of curves, so we shall get a differential equation of first order.

Differentiating (i) with respect to x, we get

2c(y + c) $$dy\over dx$$ = $$3x^2$$                          ………(ii)

Dividing (i) by (ii), we get

$$c(c + y)^2\over 2c(y + c){dy\over dx}$$ = $$x^3\over 3x^2$$

$$\implies$$ y + c = $$2x\over 3$$$$dy\over dx$$

$$\implies$$ c = $$2x\over 3$$$$dy\over dx$$ – y

Substituting the value of c in (i), we get

($$2x\over 3$$$$dy\over dx$$ – y)$$({2x\over 3}{dy\over dx})^2$$ = $$x^3$$

$$\implies$$ $$8\over 27$$$$x({dy\over dx})^3$$ – $$4\over 9$$$$({dy\over dx})^2$$y = x

$$\implies$$ $$8x({dy\over dx})^3$$ – $$12y({dy\over dx})^2$$ = 27x

This is the required differential equation of the curves represented by (i).

Example : Form the differential equation representing the family of curves y = A cos(x + B), where A and B are parameter.

Solution : The equation of the family of curves is y = A cos(x + B)                            ……….(i)

This equation contains two arbitratry constants. So, let us differential it two times to obtain a differential equation of second order.

Differentiating (i) with respect to x, we get

$$dy\over dx$$ = -A sin(x + B)                          ………(ii)

Differentiating (ii) with respect to x, we get

$$d^2y\over dx^2$$ = -A cos(x + B)

$$\implies$$  $$d^2y\over dx^2$$ = -y             [Using (i)]

$$\implies$$  $$d^2y\over dx^2$$ + y  = 0, which is the required differential equation of the given family of curves.