Differential equations

1. Differential Equations of
first order and higher degree
(Ordinary Differential Equations (ODE))
MATHEMATICS-II (20MA103T)

2. PRELIMINARIES
• Equations which are composed of an unknown function and its
derivatives are called differential equations.
• Differential equations play a fundamental role in engineering
because many physical phenomena are best formulated
mathematically in terms of their rate of change.
• For example, Newton’s second law with drag is given by:
v (velocity)- dependent variable
t (time)- independent variable
m (mass), g (acceleration due to gravity) and c
(drag co-efficient) are constants

3. • When a function involves one independent variable, the equation is
called an ordinary differential equation (or ODE).
• Examples:
In each of the above examples, x is an independent and y is a dependent variable.

4. A partial differential equation (or PDE) involves two or more
independent variables.
Examples:
Here, u is dependent variable and x and y are independent variables.
In both these examples, u is dependent variable and x and t are
independent variables.

5. Order of Differential Equation
The order of the differential equation is order of the highest derivative in the
differential equation.
Differential Equation ORDER
1
2
3
Thus, differential equations are also classified as per their orders:
a first order equation includes a first derivative as its highest derivative,
a second order equation includes a second derivative and so on.

6. Degree of Differential Equation
The degree of a differential equation is power of the highest order derivative
term in the differential equation.
Differential Equation DEGREE
1
1
3

7. Linear Differential Equation:
A differential equation is linear, if
1. every dependent variable and its derivatives are of degree one,
2. no product of dependent variables and/or their derivatives occur.
Example: 2
is non - linear because the 2nd term is not of degree one.
Example: 1 is linear.
Example: 3
is non - linear because the 2nd term involves the product of y and dy/dx.
Example: 4
is non - linear because sin y is non-linear.

8. CLASSIFY THE FOLLOWING DIFFERENTIAL EQUATIONS

9. • 1st order
• Linear
• Nonhomogeneous
• Initial value problem
(IVP)
More examples
• 2nd order
• Linear
• Nonhomogeneous
• Boundary value
problem (BVP)
• 2nd order
• Linear
• Homogeneous
• Initial value problem
(IVP)
• 2nd order
• Nonlinear
• Homogeneous
• Initial value problem
(IVP)

10. Solution of Differential Equation
Any relation between the dependent and independent variables, when substituted
in the differential equation reduces it to an identity is called a solution or an
integral of the given differential equation.
For example, y=ce2x is a solution of y’=2y, where ‘c’ is a constant.
Since the solution is true for all values of c, it is also termed as a general
solution of the given differential equation.
When the arbitrary constant of the general solution takes some unique value,
then the solution becomes the particular solution of the equation.

11. METHODS OF SOLVING FIRST ORDER
FIRST DEGREE EQUATIONS

15. DIFFERENTIAL EQUATIONS OF FIRST ORDER
AND HIGHER DEGREE
In this section, we shall discuss the set of all first order higher degree ODEs.
These include:
1. Equations solvable for p (i.e. dy/dx)
2. Equations solvable for y
3. Equations solvable for x
4. Equations homogeneous in x and y
5. Clairaut equation
6. Lagrange equation

21. Putting this value of p in the given ODE (1), we get:

22. 4. Clairaut’s Equation
Hint: Replace p by ‘a’.