 # Differential Equations

A differential equation is an equation which contains one or more terms which involve the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable)

dy/dx = f(x)

Here “x” is an independent variable and “y” is a dependent variable

For example, dy/dx = 5x

A differential equation that contains derivatives which are either partial derivatives or ordinary derivatives. The derivatives represent a rate of change, and the differential equation describes a relationship between the quantity that is continuously varying and the speed of change. There are a lot of differential equations formulas to the solution of the derivatives.

## Types

Differential equations can be divided into several types namely

• Ordinary Differential Equations
• Partial Differential Equations
• Linear Differential Equations
• Non-linear differential equations
• Homogeneous Differential Equations
• Non-homogenous Differential Equations

## Differential Equations Solutions

To find the solution of differential equation, there are two methods to solve differential function.

1. Separation of variables
2. Integrating factor

Separation of the variable is done when the differential equation can be written in the form of dy/dx= f(y)g(x) where f is the function of y only and g is the function of x only. Taking an initial condition We rewrite this problem as 1/f(y)dy= g(x)dx and then integrate them from both sides.

Integrating factor technique is used when the differential equation is of the form dy/dx+p(x)y=q(x) where p and q are both the functions of x only.

First-order differential equation is of the form y’+ P(x)y = Q(x). where p and q are both functions of x and hence called first-order differential equation because it contains functions and the first derivative of y. The higher-order differential equation is an equation that contains derivatives of an unknown function which can be either a partial or ordinary derivative. It can be represented in any order.

We also provide differential equation solver to find the solutions for related problems.

### Order of Differential Equation

The order of the differential equation is the order of the highest order derivative present in the equation. Here some of the examples for different orders of the differential equation are given.

• dy/dx = 3x + 2 , The order of the equation is 1
• (d2y/dx2)+ 2 (dy/dx)+y = 0. The order is 2
• (dy/dt)+y = kt. The order is 1.

First Order Differential Equation

You can see in the first example, it is a first-order differential equation which has degree equal to 1. All the linear equations in the form of derivatives are in the first order.  It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as;

dy/dx = f(x,y) = y’

Second-Order Differential Equation

The equation which includes second-order derivative is the second-order differential equation.  It is represented as;

d/dx(dy/dx) = d2y/dx2 = f”(x) = y”

### Degree of Differential Equation

The degree of the differential equation is the order of the highest order derivative, where the original equation is represented in the form of a polynomial equation in derivatives such as y’,y”, y”’, and so on.

Suppose (d2y/dx2)+ 2 (dy/dx)+y = 0 is a differential equation, so the degree of this equation here is 2. See some more examples here:

• dy/dx + 1 = 0, degree is 1
• y”’ + 3y” + 6y’ – 12 = 0, degree is 3

### Ordinary Differential Equation

An ordinary differential equation involves function and its derivatives. It contains only one independent variable and one or more of its derivative with respect to the variable.

The order of ordinary differential equations is defined to be the order of the highest derivative that occurs in the equation. The general form of n-th order ODE is given as

F(x, y,y’,….,yn ) = 0

## Applications

Let us see some differential equation applications in real-time.

1) Differential equations describe various exponential growths and decays.

2) They are also used to describe the change in investment return over time.

3) They are used in the field of medicines for modelling cancer growth or the spread of disease in the body.

4) Movement of electricity can also be described with the help of it.

5) They help economists in finding optimum investment strategies.

6) The motion of waves or a pendulum can also be described using these equations.

The various other applications in engineering is: ­ heat conduction analysis, in physics it can be used to understand the motion of waves, pendulums, in chemistry it is used for modelling the chemical reactions, in medical science for monitoring the cancer growth. The ordinary differential equation can be utilized as an application in engineering field like for finding the relationship between various parts of the bridge.

### Linear Differential Equations Real World Example

To understand Differential equations, let us consider this simple example. Have you ever thought why a hot cup of coffee cools down when kept under normal conditions? According to Newton, cooling of a hot body is proportional to the temperature difference between its own temperature T and the temperature Tof its surrounding. This statement in terms of mathematics can be written as:

dT/dt ∝ (T – T0)…………(1)

This is the form of a linear differential equation.

Introducing a proportionality constant k, the above equation can be written as:

dT/dt = k(T – T0)   …………(2)

Here, T is the temperature of the body and t is the time,

T0 is the temperature of the surrounding,

dT/dt is the rate of cooling of the body Fig: The path of the projectile follows a curve which can be derived from an ordinary differential equation. Eg:­ dy/dx = 3x

Here, the differential equation contains a derivative that involves a variable (dependent variable,y) w.r.t another variable(independent variable,x). The types of differential equations are ­:

1. An ordinary differential equation ­contain one independent variable and its derivatives. It is frequently called as ODE. The general definition of the ordinary differential equation is of the form:­ Given an F, a function os x and y and derivative of y, we have

F(x, y, y’ …..y^(n­1)) = y (n) is an explicit ordinary differential equation of order n.

2. Partial differential equation ­that contains one or more independent variable.

### Example

Question:

Verify that the function y = e-3x is a solution to the differential equation $\frac{d^2y}{dx^2}~ + ~\frac{dy}{dx} ~-~ 6y$ = $0$.

Solution:

The function given is $y$ = $e^{-3x}$. We differentiate both the sides of the equation with respect to $x$,

$\frac{dy}{dx}$ = $- 3 e^{-3x}$   …………(1)

Now we again differentiate the above equation with respect to x,

$\frac{d^2y}{dx^2}$ = $9 e^{-3x}$ …………(2)

We substitute the values of $\frac{dy}{dx}, \frac{d^2y}{dx^2}$ and $y$ in the differential equation given in the question,

On left hand side we get, LHS = 9e-3x + (-3e-3x)-6e-3x

= 9e-3x – 9e-3x = 0 (which is equal to RHS)

Therefore the given function is a solution to the given differential equation.

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