Education Journalist | Study Abroad Lead
First Order Differential Equation is an equation of the form f (x,y) = dy/dx where x and y are the two variables and f (x,y) is the function of the equation defined on a specific region of a x-y plane. A differential equation is mostly used in subjects like physics, engineering, biology and chemistry to determine the function over its domain and some derivatives. In this article, we will understand differential equations, first order differential equations, types of first order differential equations and methods to solve the first order differential equations.
Also Read: NCERT Solutions For Class 12 Mathematics Chapter 10 Differential Equation
Key Terms: Differential equation, variable, Integrating factor method, Costant factor method, function, derivative, First Order Differential Equation.
What is a Differential Equation?
[Click Here for Sample Questions]
A differential equation is a mathematical expression that comprises one or more functions and their derivatives. The function's derivatives define the rate of change of a function at a given moment. It is mostly used in subjects like physics, engineering, and biology. The fundamental goal of differential equations is to find the solutions that fulfil the equations as well as the properties of the solutions.
The term derivative is the root of the first order differential equation. A derivative is a mathematical technique used to calculate the rate of change of values in a function at a certain point in the function. It is also known as a slope, which denotes the ratio of the rate of departure in the value of a certain function depending on an independent variable.
Also Read: Applications of Derivatives
First Order Differential Equation
[Click Here for Sample Questions]
The first order differential equation is determined by the equation dy/dx =f (x,y) with two variables x and y and function f(x,y) specified on a xy-plane area. Since it only has the first derivative dy/dx, the equation is of first order, and no higher-order derivatives exist.
The first-order differential equation can alternatively be expressed as:
y’ + a (x)y = f (x)
or
dydt= f (y,t)
In general, the differential equation is used to express a relationship between a function and its derivatives. It is used in physics and chemistry to determine the functions across its domain if we know the functions and some of their derivatives.
Also Read:
| Related Articles | ||
|---|---|---|
| Definite Integral Formula | Limit Formula | Tan2x Formula |
Types of First Order Differential Equations
[Click Here for Sample Questions]
There are five major forms of differential equations. They are as follows:
- Linear Differential Equations-
A linear differential equation is an equation that contains a variable, its derivative and a few additional functions.
A linear differential equation has the conventional form dy/dx + Py = Q, and it contains the variable y and its derivatives.
- Homogeneous Equations-
If the degree of f(x,y) and g(x,y) is the same, a differential equation of the form f(x,y)dy = g(x,y)dx is said to be homogeneous.
A homogeneous function of degree n, for k0, is a function of the type F(x,y) that may be represented as kn F(x,y).
- Exact Equations-
The formula P (x,y) dx + Q (x,y) dy=0 is an exact differential equation if and only if there exists a function f of two variables x and y with continuous partial derivatives that separates the exact differential equation formulation.
Also Read: Methods of Integration
- Separable Equations-
Any equation that can be expressed in the form y′=f(x)g is a separable differential equation (y).
To obtain the general solution to a separable differential equation, the method of variable separation is applied.
- Integrating Factor-
The integrating factor is defined as the function chosen to solve the given differential equation.
It is mostly used in first-order ordinary linear differential equations. P(x) is a multiple of y and signifies the integrating factor.
dydx+ p (x) y(x) = q (x)
Also Read: Differentiation and Integration Formula
Methods To Solve First Order Differential Equations
[Click Here for Sample Questions]
There are two methods for solving first-order linear differential equations:
- Integrating factor method
- Constant variation method.
Let’s discuss more about these two methods in detail:
-
Integrating Factor
Express linear differential equation is expressed in its standard form:
y’ + a (x)y = f(x)
The formula below defines the integrating factor:
\(u(x) = exp (\int a (x) dx)\)
When the left side of the equation is multiplied by the integrating factor u(x), the left side is converted into the derivative of the product y(x) u(x).
The differential equation's general solution is written as follows:
\(y = \frac{\int u (x) f (x) dx + C}{u(x)}\)
where ‘C’ is a constant.
-
Variation Of Constant
First, find the general solution of the homogeneous equation:
y’ + a (x)y = 0
A constant of integration 'C' appears in the general solution of the homogeneous equation.
We can replace the constant 'C' with an unknown function C(x). We can get the function C by substituting this solution into the non-homogeneous differential equation (x). The procedure presented here is known as the method of variation of constant.
Also Read: homogeneous function
Properties of First Order Differential Equation
[Click Here for Sample Questions]
First order differential equations have the following characteristics:
- The function y and its derivatives appear in the equation only up to the first degree.
- There are no y products and/or derivatives present.
- There are no transcendental functions (trigonometric, logarithmic, etc.) of y or its derivatives.
Applications of First Order Differential Equations
[Click Here for Sample Questions]
The following are the applications of First Order Differential Equations:
- To Solve Orthogonal Trajectories
- To find growth and decay at molecular level
- Newton's Law of Cooling
- To solve dilution and falling body problems
- Electrical Circuits
Things to Remember
- A mathematical expression which consists of one or more functions and their derivatives is called a differential equation.
- First Order Differential Equation is an equation of the form f (x,y) = dy/dx where x and y are the two variables and f (x,y) is the function of the equation.
- The general form of a linear differential equation is represented by- dy/dx + Py =Q, and it contains the variable y and its derivatives.
- A homogeneous differential equation is represented by the form f(x,y)dy = g(x,y)dx only if the degree of f(x,y) and g(x,y) is the same,
- First order differential equations can be solved by two methods: Integrating factor and variation of constant.
Also Read:
Sample Questions
Ques: What is the first-degree equation in one variable? (2 Marks)
Ans: First degree equations in one variable are algebraic equations in which each term is either a constant or the product of a constant and the first power of a single variable. Linear equations appear often in real-world problems.
Ques: What is differential calculus? (2 Marks)
Ans: Differential calculus is a branch of mathematics that analyses the rates at which quantities change. The derivative of a function at a certain input value describes the function's rate of change at that input value. Differentiation refers to the process of determining a derivative.
Ques: Find the differential equation for y = x -A/x ? (3 Marks)
Ans: Given that:
\(y = x - \frac{A}{x}\)
\(\frac{dy}{dx} = 1 + \frac{A}{x^2}\)
Try to eliminate A by,
- Divide 1st Equation with ‘x’
- Calculate the 2nd & 3rd Equation
\(\frac{y}{x}= 1 - \frac{A}{x^2}\)
\(\frac{dy}{dx} + \frac{y}{x} = 2\)
Ques: Form a suitable differential equation using y = A cos x + B sin x (3 Marks)
Ans: Given that:
y = A cos x + B sin x
On differentiating, we get
\(y' \) = – A sinx + B cos x
\(y''\) = – A cos x – B sin x
= – (A cos x + B sin x)
\(y'' = - y \implies \frac{ d^2y}{dx^2} + y = 0\)
Ques: Find the general solution of y’ + 2x y = x. (3 Marks)
Ans: Calculate
\(h(x) = \int 2x dx = x^2 and e^{h(x)} =e^{x^2}\)
Multiply by ex2
On integrating, we get
Ques: y'+2xy = x is a differential equation. Find the general equation of the given equation. (3 Marks)
Ans: The given equation is already in standard form (x) i.e. y' + P(x)y = Q
As a result, P(x) = 2x and Q(x) = x.
Now multiplying both side by;
\(\mu(x) = \int_e^{Pdx} = \int_e^{2xdx} =e^{x^2}\)
\(e^{x^2} + 2 x e^{x^2}y = xe^{x^2}\)
\(\frac{d}{dx}(e^{x^2}y) =xe^{x^2}\)
Now integrating both side, we get;
\(e^{x^2} y = \int xe^{x^2}dx\)
\(e^{x^2} = \frac{1}{2} e^{x^2} + c\)
\(y = \frac{1}{2} + ce^{-x^2}\)
For Latest Updates on Upcoming Board Exams, Click Here: https://t.me/class_10_12_board_updates
Check-Out:
Comments