Page 110 - DMTH202_BASIC_MATHEMATICS_II

Unit 8: Formation of Differential Equation

Notes
2
2
2
 v   v 
(ii) 2  C   2  
t  x 
v v v
x  y  z  nz
(iii)
x y z
are partial differential equations.

Self Assessment

Fill in the blanks:
5. A differential equation can be classified as an ordinary or ................................ differential
equation.
6. A differential equation involving a single independent variable and the derivatives with
respect to it, is called an ................................ differential equation.
7. A differential equation involving ................................ independent variables and the partial
derivatives with respect to them, is called a partial differential equation.
8.3 Formation of a Differential Equation

At times a family of curves can be displayed by a single equation. In this case the equation
includes an arbitrary constant c. By allocating different values for c, we obtain a family of
curves. Here c is known as the parameter or arbitrary constant of the family.
Differential equations are formed by elimination of arbitrary constants. To eliminate two
arbitrary constants, we require two more equations besides the given relation, leading us to
second order derivatives and hence a differential equation of the second order. Elimination of n
arbitrary constants leads us to n order derivatives and hence a differential equation of the
th
th
n order.

Notes By eliminating the arbitrary constants from the specified equation and the equations
attained by the differentiation, we obtain the requisite differential equations.

Example: From the differential equation of all circles of radius r.

Solution:
The equation of any circle of radius r is
2
2
2
(x – h) + (y – k) =r , ………..(1)
where (h, k) the coordinates of the centre.
Differentiating (1) w.r.t. x, we get
dy
2(x – h) + 2(y – k) = 0
dx

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