Page 115 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes 8.4 Summary

 A differential equation involves independent variables, dependent variables, their
derivatives and constants.

 A differential equation involving a single independent variable and the derivatives with
respect to it, is called an ordinary differential equation.

 An ordinary differential equation contains total derivatives or total differentials.
 A differential equation involving two or more independent variables and the partial
derivatives with respect to them, is called a partial differential equation.

 By allocating different values for c, we obtain a family of curves where c is known as the
parameter or arbitrary constant of the family.
 Differential equations are formed by elimination of arbitrary constants.
th
 Elimination of n arbitrary constants leads us to n order derivatives and hence a differential
th
equation of the n order.
 By eliminating the arbitrary constants from a particular equation and the equations
achieved by the differentiation, we obtain the requisite differential equations.

8.5 Keywords

Arbitrary Constant: By allocating different values for c, we obtain a family of curves where c is
known as the parameter or arbitrary constant of the family.
Differential Equation: A differential equation involves independent variables, dependent
variables, their derivatives and constants.

Ordinary Differential Equation: A differential equation involving a single independent variable
and the derivatives with respect to it, is called an ordinary differential equation.
Partial Differential Equation: A differential equation involving two or more independent
variables and the partial derivatives with respect to them, is called a partial differential equation.

8.6 Review Questions

1. Form the differential equation for y = a cos x + b sin 3x where a and b are arbitrary
3
constants.
bx
2. Form the differential equation of y = ae where a and b are the arbitrary constants.
2
2
3. Find the differential equation for the family of concentric circles x + y = a , a is the arbitrary
2
constant.
4. Obtain the differential equation of the family of circles with fixed radius r and center on
the y-axis.
5. Form the differential equation of all circles with their centers on the line y = 2x.


6. Form the differential equation of simple harmonic motion given by x = A cos (nt ) .
7. Obtain the differential equation from the relation
3x
2x
i. y = C e + C e + C e x
1 2 3

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