Differential Equations: Overview, Questions, Preparation

Calculus is an advanced mathematical topic generally taught to students during the final years of their schooling. Calculus is extremely important if you are looking to delve into the fields of pure sciences or engineering and gain further knowledge about small scale and large scale phenomena. Among the initial concepts taught to students in their journey through calculus is differential equations.

Any equation that involves an independent variable, a dependent variable, a constant and the derivative of the dependent variable with respect to the independent variable is known as a differential equation. The general form of a differential equation is as follows.

xdy/dx+xyd²y/dx²+c=0, dz/dx+dz/dy=0

Order and Degree of Differential Equations

A differential equation’s order is always a positive integer and is the highest occurring derivative. A differential equation’s degree is the exponent of the derivative that has the highest order.

There are a number of terms and definitions that are associated with differential equations.  

Order of Differential Equation

The order of a differential equation is the order of the term in the equation, in which the derivative of the dependent variable with respect to the independent variable has been taken the greatest number of times. This number is also the order of that term. The order of a differential equation can never be greater than the number of arbitrary constants in the equation's general form.

Differential equations comprise an independent variable, a dependent variable, and a differential coefficient.

The differential coefficient is of the dependent variable and varies with the independent variable.

For example, x² (d²y / dx²) + x³ (dy / dx)³ 7 x² y².

Generally, the order of the differential equation is confused with the degree of the differential equation. The differential equation's degree is the greatest power of the highest order term in the differential equation.

Linear and Nonlinear Differential Equations

A linear differential equation has one or more dependent variables and its derivatives in the first power. For example, a linear equation of the n^th order is:

P₀  dⁿy / dx_n + P₁ d^n–1 y / dx^n-1 + P₂ d^n-2 y / dx^n-2 and so on.
Where P₀, P₁, P₂, and so on are constants or functions of x.

Differential Equations and Their Solutions

The solution of a differential equation can be of two types - the general solution and the particular solution. The differential equation's general solution basically lays out the standard format of the solution and contains arbitrary constants. For a particular solution, the value of the solution at two points is required. By inserting these values, the values of the arbitrary constants can be ascertained, providing a specific solution.

To find the solution to a differential equation, we must establish a relationship between the variables such that there are no differential coefficients and the derivation satisfies the equation.

For example, if d2y / dx2 + y = 0, is a differential equation, then, by integrating it, we get

Y = A cos x + B sin x.

There are two solutions to a differential equation.

General Solution

When we solve the differential equation and get as many constants as in the order of the equation,
then we call such a solution as a general solution or a complete integral.

In the above example, d²y / dx² + y = 0, by integrating the equation, we got, Y = A cos x + B sin x. So, this is the general solution to a differential equation.

Particular Solution

If the arbitrary constants have particular values, then you have a particular solution to the differential equation.

The NCERT textbook pays a lot of heed to the chapter on differential equations. Class XI and Class XII NCERT Mathematics are significantly about calculus, and the textbooks do a great job of explaining each concept of calculus to the students. A number of solved examples are provided which help students gain essential practice in solving simple differential equations. From the examination point of view as well, differential equations are important. They can form 1 mark, 2 mark, and 4 mark questions in Class XI, which may also be lengthy to solve if steps are required to be demonstrated.

The chapter, ‘Differential Equations’ is a part of the unit, ‘Calculus’, prescribed for CBSE class 12. The chapter carries 44 marks in the exam and is an important one.

1. What is the general solution of the equation dz/dy+ 5y²=1?
Solution.

The general solution of the equation is

dz = -5y²dy + dy

So, z = -5y³/3 + y + c

2. What is the specific solution of the equation dz/dy+ y=1; z = 0, y = 1?
Solution.

The specific solution of the equation is
dz = -ydy + dy
So, z = -y²/2 + y + c
When z = 0, y = 1
So, c = -1/2
Hence, z = -y²/2 + y -1/2

3.What is the order and degree of the equation d²z/dy²+ y=1?

Solution.

The order of the equation is 2 and the degree of the equation is 1.

4. Solve the following differential equation.
Dv / dx + (1 + y²) / y = 0.

Solution:

Given, dv / dx + (1+y²) / y = 0.
Or dv / dx = - (1+y²) / y.
Or y / (1+y²)dy = - dx.
Now, 2y / (1+y²)dy = - 2dx
Or log (1+y²) = -2x + c.
So, we have the solution to the differential equation to be ½ log (1+y²) + x = c. 

5. Solve the following differential equation.
Dy/dx = (1 + y²) / y³

Solution:

Given that, dy / dx = (1 + y²) / y³.
So, y³ / (1 + y²) dy = dx.
Or, [y–y / (1 + y²)] dy = dx.
Ydy–½ 2y / (1 + y²)dy = dx.
Or, y² / 2–½ log (y² + 1) = x + c.
Thus, we have the solution to the differential equation to be y² / 2 – ½ log (y² + 1) = x + c.

6. Solve the differential equation, dy / dx = sin² y.

Solution:
Given that, dy / dx = sin² y.
Or dy / sin² y = dx.

Now, we know that 1 / sin x = cosec x.
So, cosec² x dy = dx.

By integrating both sides, we get - cot y = x + c.

Thus, the solution for the given differential equation is – cot y = x + c.

Q: What is a Differential Equation?

Q: What are the two types of differential equation solutions?

Q: Which type of solution has arbitrary constants?

Q: What is the degree of a differential equation?

Q: What are the Order and Degree of a Differential Equation?

Q: What is a Linear Differential Equation?

Q: What are the different ways in which you can Solve a Differential Equation?

Q: What are Homogeneous Differential Equations?

Q: What do you mean by the standard form of a differential equation?