Mathematics - Differential Equations

1.  If y is a function of x, then we denote it as
y = f(x). Here x is called an independent
variable and y is called a dependent variable.
 If there is a equation dy/dx = g(x) ,then this
equation contains the variable x and
derivative of y w.r.t x. This type of an
equation is known as a Differential Equation.

2.  Order of the highest order derivative of the
dependent variable with respect to the
independent variable occurring in a given
differential equation is called the order of
differential equation.
 E.g. – 1st order equation
 2nd order equation

3.  When a differential equation is in a
polynomial form in derivatives, the highest
power of the highest order derivative
occuring in the differential equation is called
the degree of the differential equation.
 E.g. – Degree – 1 ,(d²y/dx) + dy/dx = 0
Degree – 2 , (d²y/dx)² + dy/dx = 0

4. 1. Ordinary Differential Equation - An Ordinary
Differential Equation is a differential
equation that depends on only one
independent variable. E.g. – dy/dt = k(y)t is
an Ordinary Differential Equation because
y(the independent variable) depends only on
t(the independent variable).

5. 2 . Partial Differential Equation - A Partial
Differential Equation is differential equation
in which the dependent variable depends on
two or more independent variables.
E.g. – d²f/dx² + d²f/dy² = 0 is a Partial
Differential Equation because f depends on
two independent variables x and y.

6. 3 . Linear Differential Equation - A first-order
differential equation is linear if it can be
written in the form dy/dt + g(t)y = r(t) where
g(t) and r(t) are arbitrary functions of t.
E.g. – dy/dt = t²y + cost(t) is a first-order
linear differential equation where g(t) = t²
and r(t) = cos(t)

7. 4 . Nonlinear Differential Equation -
It is a differential equation whose right hand
side is not a linear function of the dependent
variable.
E.g. -

8. 5 . Homogeneous Differential Equation(Same
Degree) - A linear first-order differential
equation is homogeneous if its right hand
side is zero , that is r(t) = 0
E.g. -

9. 6 . Non homogeneous Differential Equation - A
linear first-order differential equation is non
homogeneous if its right-hand side is non-
zero that is r(t) ≠ 0
E.g. -

10.  If for a function y = f(x), defined on some
interval ,there exist derivatives of up to order
n and if the function f and its derivative
together satisfy the given differential
equation , then y = f(x) is called a solution of
differential equation.

11. There are 3 type of solutions of Differential
Equation.
1. General solution – there are many constants
we need not need to find the value of them.
2. Particular solution – there are many
constants and we need to find value of them.
3. Singular solution – if the solution can not be
found out through general and particular
solution.

12.  Solution of Differential Equation of first order
& degree can be found out through Method
of Variable and Separable.

13.  The study of Differential equation began in
order to solve the problems that originated
from different branches of
mathematics,physics,biological sciences etc
 Application of Differential equations are in
following fields -:
1. Physics (RL Circuit)
2. Applications in Geometry

14. 3. Exponential growth
4. Exponential decay
5. Newton's law of cooling

15.  Let us take two examples on applications of
differential equations,
1. Application in Geometry
2. Exponential growth