Differential Calculus: Overview, Questions, Preparation

Calculus is the branch of mathematics that deals with the rate of change of a function or continuous change of a function. It is a study of how things change, over a small interval of time. It has two main branches which include differential calculus and integral calculus.
Differential calculus involves the rate of change of a function to other variables in an infinitesimally small area.

Representation

The representation of a derivative is by dy/dx or f’(x), which is a derivative of y with respect to variable x.

The solution obtained from the general solution by giving particular values to the arbitrary constants is called a particular solution of the differential equation.

The chapter about differential equations is a part of the class 12 mathematics curriculum. A good understanding of concepts of calculus and derivatives is essential to understand and solve the questions related to differential calculus.

This chapter holds a weightage of around 10 marks in the mathematics of class 12 board exams, out of the total 35 marks dedicated for calculus.

1.Verify that the function y = e– 3x is a solution of the differential equation 
d²y/dx²+dy/dx-6y= 0
Solution:
The function is y = e– 3x. both sides of the equation when differentiated with respect to x, we get
dy/dx = −3e^-3x ... (1) 

Now, differentiating the first equation with respect to x, we have 

d²y/dx² = 9 e^-3x

Substituting the values of 

d²y/dx², dy/dx,  and y in the earlier differential equation, we get 

L.H.S. = 9e^-3x + (–3e^-3x ) – 6.e^-3x = 9e^-3x – 9e^-3x = 0 = R.H.S.. 

Therefore, the given function is a solution of the above differential equation.

2. Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constants.
Solution.
y = a sin(x + b) ... (1)
Differentiating both sides of equation (1) with respect to x, successively we get
dy/dx = a cos (x + b) ... (2)
d²y/dx² = – a sin(x + b) ... (3)
Eliminating a and b from equations (1), (2) and (3), we get 
d²y/dx² + y = 0 ... (4)

This is free from arbitrary constants a and b, and is hence the required differential equation.

3. Form the differential equation representing the family of curves y = mx, where, m is an arbitrary constant
Solution.
According to the question,
y = mx ... (1)
Differentiating both sides of equation (1) concerning x, we get
dy/dx= m
Substituting the value of m in equation (1) we get
y =(dy/dx)x 
or
x dy/dx - y = 0
which is free from the parameter m and is hence the required differential equation.

Q: What are the applications of differential calculus?

Q: What is an order of a differential equation?

Q: What is the degree of a differential equation?

Q: What is a differential equation used for?

Q: What is a solution to a differential equation?