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.