Differential Equations: Degree, Types and Orders

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 following are various types of differential equations:

1. First Order Differential Equations

This is a one-derivative differential equation that has no higher-order derivatives. Here, we are solving y(x), which is the function whose rate of change will obey the rule. Since there is only one derivative, integration will lead to a single constant, c.

The canonical form of the equation is:

$$\frac{d}{\mathrm{dx}}y=f(x,y).$$

2. Second-order derivative

The second-order derivative tells you how the slope itself is changing. 

a(x)y′′ + b(x)y′ + c(x)y = g(x).

y′′ means that you will be picking up two constants of integration.

The degree of a differential equation represents the power of the highest order derivative. Here, the original equation is in the form of a polynomial equation in derivatives like y’,y” and y”. Let us consider an equation to determine its degree. 

$$\frac{d^{2}y}{d^x}+2\frac{dy}{dx}+y=0$$ The degree of the differential equation, in this case, will be 1. 

The following are the different types of differential equations:

• Homogeneous Differential Equations: If the differential equation of form f(x,y) dy = g(x,y) dx is considered to be homogeneous when the degree of f(x,y) and g(x,y) is same.

$$\frac{dy}{dx}=\frac{x+y}{x-y}$$

• Nonhomogeneous Differential Equations: It is a linear differential equation where an external "forcing" term appears on right-hand side.

$$\frac{dy}{dx}+2y=e^x$$

• Linear Differential Equations: This type of equation consists of dependent variable's derivatives w.r.t. one or more independent variables.

$$\frac{d^{2}y}{d^x}-4\frac{dy}{dx}+3y=0$$

• Nonlinear differential equations: Here, an unknown function y(x) and its derivatives appear nonlinearly.

$$\frac{dy}{dt}=ry(1-\frac{y}{K})$$

• Ordinary Differential Equations: This type of differential equation has a function and its derivatives. It includes a single independent variable along with one or more of derivatives with respect to variable.

$$\frac{d^{2}y}{d^x}+y=0$$

• Partial Differential Equations: It contains the partial derivatives of one or more dependent variables with more than 1 independent variable.

$$\frac{\partial u}{\partial t}=k\frac{{\partial }^{2}u}{{\partial }^x}$$

You can use this differential equation calculator for practice before exams. This type of calculator can help you calculate any kind of differential equation based on Euler's method and Runge-Kutta 4th order. Students will be required to mention the:

1. differential equation
2. Initial Value (y₀):
3. Initial x (x₀):
4. Final x 
5. Number of steps

Based on your inputs, the calculator will provide you with the numerical solutions of the equation. 

Differential equations are important for JEE Main and IIT JAM aspirants. By practising sample questions, they can understand the complexity of the questions asked in such entrance exams:

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.