The basic concepts of differential equations are:

I. Order of a Differential Equation

The derivative with the highest order in an equation becomes the order of a differential equation. Here are some examples for different orders of differential equations:

• In the differential equation, dy/dx = 7x + 3, the order of the equation is 1.
• In the differential equation (d2y/dx2) + 5(dy/dx) – 3y = 0, the order of the equation is 2.

If the order of the equation is 1, it is a first order differential equation such as in the first example. It is represented as dy/dx = f(x, y) = y’

If the order of the equation is 2, it is a second order differential equation such as in the second example. It is represented as d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.

II. Degree of a Differential Equation

In a differential equation, the power of the highest order derivative becomes the degree of the differential equation. Here are some examples:

• In the differential equation (d2y/dx2) + 3 (dy/dx) + y = 0, the degree of the equation is 1.
• In the differential equation, dy/dx + 20 = 0, the degree of the equation is 1.

In the differential equation (y”’)4 – 6y” + 7y’ + 10 = 0, the degree of the equation is 3.

When the solution of a differential equation contains arbitrary constants, the solution is called a general solution of the differential equation.

When unique values are assigned to the arbitrary constants derived from the general solution, the solution becomes a particular solution of the differential equation.

For example, if dy/dx = sin2x + ex + 3x4

Then the general solution is: y = cos(2x/2) + ex + x5/3 + C

Now, substituting x = 0, y = 3 in the general solution: 3 = cos(0)/2 + e0 + (0)5/3 + C

C = 2

Therefore, if C = 2 is substituted in the general solution, then y = cos(2x/2) + x5/3 + 2, which is the particular solution of the differential equation.

The solution for any given differential equation is of the form f(x, y, a1,a2,…..,an) = 0, where x and y are the variables and a1 , a2 …...an are the arbitrary constants. Here are the steps to be followed in the formation of a differential equation:

Step I: Note down the equation and find the number of arbitrary constants it has.

Step II: The equation is differentiated with respect to the dependent variable n times, where n is the number of arbitrary constants.

Step III: The arbitrary constants are then eliminated from the equations obtained in step II.

Step IV: After elimination of the arbitrary constants, the equation that remains is the differential equation required.

First-order, first-degree differential equations can be solved using the following three methods:

a. Variable separable form

​​Take the differential equation dy/dx = F(x, y).

To begin with, separate the variables. Following this, integrate both sides to get the general solution:

dy/dx = h(x)・k(y)

After separating the variables, we get: dy/k(y) = h(x) dx.

Next, integrate the above equation to get the general solution:

K(y) = H(x) + C

Here, K(y) and H(x) are the anti-derivatives of 1/K(y) and h(x) respectively while C is the arbitrary constant.

b. Homogeneous differential equations

A differential equation dy/dx = f(x, y)/g(x, y) is homogeneous if f(x, y) and g(x, y) are homogeneous functions of same degree, i.e. it may be written as

dy/dx = xnf(y/x)/xng(y/x) = f(y/x)/g(y/x) =F(y/x)...........(i)

To find out whether the differential equation is homogeneous or not, it is written as dy/dx = F(x, y) or dx/dy = F(x, y), and replace x by λx, y by λy to write F(x, y) = λ F(x, y).

The differential equation is homogeneous if the power of λ is zero.

Next, we put y = vx ⇒ dy/dx = v + x dv/dx in equation (i) to reduce it into variable separable form. Then, solve it and put v = y/x to get the required solution.

It is important to remember that if the homogeneous differential equation is in the form of dy/dx = F(x, y), where F(x, y) is homogeneous function of degree zero, then we make substitution x/y = v, i.e. x = vy and we proceed further to find the general solution as mentioned above.

c. Linear differential equations

General form of a linear differential equation is

dy/dx + Ay = B …(i)

where, A and B are functions of x or constants.

OR

dx/dy + A’x = B’ …(ii)

where, A’ and B’ are functions of y or constants.

Then, the solution of equation (i) is given by the equation y × IF = ∫(B × IF) dx + K

where, IF = Integrating factor and IF = e∫Adx

Solution of equation (ii) is given by the equation x × IF = ∫(B’ × IF) dy + K

where, IF = Integrating factor and IF = e∫A’dy

a. How do you define differential equation?

A differential equation is an equation that has the derivatives of a function, and is denoted by dy/dx. A differential equation can have derivatives with one or more dependent variables in relation to one or more independent variables.

How many types of differential equations are there?

b. Differential equations are of six types. They are as follows:

1. Ordinary Differential Equations
2. Partial Differential Equations
3. Homogeneous Differential Equations
4. Non-Homogeneous Differential Equations
5. Linear Differential Equations
6. Non-linear Differential Equations

c. Where are differential equations used in real life?

Following are a few real-life examples of when differential equations are used:

1. Differential equations are used to determine exponential growths and decays.
2. Financial experts use differential equations to determine the change in return on investment over a period of time.
3. Experts in the medical field use differential equations to study cancer growth or the spread of a disease in a patient’s body.
4. Economists use differential equations to chart out optimum investment strategies.