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 = x3 , where c is a parameter.
Solution : The equation of the family of curves is c(y+c)2 = x3 ……….(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) dydx = 3x2 ………(ii)
Dividing (i) by (ii), we get
c(c+y)22c(y+c)dydx = x33x2
⟹ y + c = 2x3dydx
⟹ c = 2x3dydx – y
Substituting the value of c in (i), we get
(2x3dydx – y)(2x3dydx)2 = x3
⟹ 827x(dydx)3 – 49(dydx)2y = x
⟹ 8x(dydx)3 – 12y(dydx)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
dydx = -A sin(x + B) ………(ii)
Differentiating (ii) with respect to x, we get
d2ydx2 = -A cos(x + B)
⟹ d2ydx2 = -y [Using (i)]
⟹ d2ydx2 + y = 0, which is the required differential equation of the given family of curves.