How to Solve First Order Differential Equations with Formula and Examples
The root of the first order differential equation lies in the term derivative. A clear understanding of derivatives can make the process of learning differential equations easier and digestible. In mathematical terms, a derivative is a tool to measure a rate of change of values in a function at a particular point in the function. It can also be understood as a slope, which signifies the ratio of the rate of deviation in the value of a particular function based on an independent variable. Now, it is a good time to deal with the question: what is a differential equation?
What is a Differential Equation?
A differential equation represents the change in the derivative of a dependent variable ‘X’ concerning an independent variable ‘Y’. A differential equation may contain more than one variable.
A differential equation is represented as
dy/dx +f(y)= Q(x)
General Solution of 1st Order Differential Equation
The general solution of the first order differential equation provides the relation between both the dependent and the independent variable devoid of any form of derivative. Also, the relation arrived at, will inadvertently satisfy the equation at hand.
We will look into the details and talk about the different approaches to solve the first order differential equation in the later stage.
First-Order Differential Equation
There are different orders of differential equations, but we will focus mainly on the first-order form. On a side note, it will come in handy if you know what is an order of a differential equation?
The order of a differential equation is always the highest order of derivatives. The order of the differential is the highest number attached to the 'd' which is in the numerator (for pictorial
understanding).
A first order differential equation indicates that such equations will be dealing with the first order of the derivative.
Again for pictorial understanding, in the first order ordinary differential equation, the highest power of 'd’ in the numerator is 1.
A first-order differential equation is one of the five different types of DE, each of them are mentioned below.
Separable equation
Integrable equation
Exact equations
Homogeneous equation
First order linear differential equation
Next, we will look into first order linear differential equations. It is one of the basic elements of DE. A proper understanding of first order linear differential equations can make the process of learning DE smooth.
First Order Linear Differential Equation
Before we head any further, it is important to familiarise yourself with the most important form of DE that is First order linear differential equation. It is the simplest form first order DE, the general solution of which can be easy to find. Thus, it is advised that students do not memorize the formulae. Next, we will look into a couple of methods to solve the 1st order differential equation.
Example - dy/dx+ R(t)y= s(t)
How to Solve 1st Order Differential Equations?
Solving 1st order differential equations can be tedious but the two approaches mentioned below can ease things out.
Method of Variation
Methods of variation of parameters deals with solving the homogeneous equation on the left-hand side to arrive at a general solution. This is done by sustaining the RHS with
zero and solving the equation accordingly. We have mentioned steps for solving 1st order differential equations.
y’+3y=g(t)
Step -1: Solving the homogeneous equation by substituting g(t)=0
The general solution of a first order differential equation arrived by solving the homogeneous equation, it will include a constant of integration, say K. The constant part can be substituted with an unknown
function K(x).
Step-2: The next step is to substitute the function K(x) into the nonhomogeneous differential equation. After the substitution, the uncertain function K(x) is determined to certainty.
Integrating Factor
For solving 1st order differential equations using integrating methods you have to adhere to the following steps.
First, arrange the given 1st order differential equation in the right order (see below)
dy/dx + A(y)= B(x)
Pick out the integrating factor, as in, IF= e ∫A(y)dx
Multiply given equation with IF. (Use the product rule to solve the equation)
Integrate both sides concerning x.
Finally, divide the equation with the IF to arrive at the final value of ‘y’.
Both methods provide the same solution of first order differential equation.
Solved First Order Differential Example
Here, we will see the first order differential equation example, so that you can have a clear idea of how to solve the first order differential equation. The below mentioned first order differential example deals with exponential function.
Question 1 : Solve the equation y′−y−xex = 0
Solution : Given, y′−y−xex = 0
First, you have to rewrite the provided equation in the form given below.,
y′−y = xex
Using the integrating factor, it becomes;
K(x)=e∫(−1)dx=e−∫dx=e−x
Now you have proceeded in the normal using the last few steps mentioned in the integrating factor method. There are various first order differential equation example on the web, which we would recommend you to have a look at.
Did You Know
Have you ever thought about the number of solutions for a differential equation? Only those who have a core understanding of the concept will know the correct answer.
A differential has an infinite number of solutions which stems from the fact that a function can have an infinite number of antiderivatives.
You can carve out DE formulae by knowing the antiderivative of a function. The formula can help you to draw the graph. This concept is used in physics for studying the motion of an oscillating object.
FAQs on First Order Differential Equations Explained
1. What is a first order differential equation?
A first order differential equation is an equation that involves the first derivative of a function but no higher-order derivatives. It is generally written as dy/dx = f(x, y).
- The highest derivative present is the first derivative.
- It relates a function y to its rate of change.
- Example: dy/dx = 3x is a first order differential equation.
2. What is the general form of a first order differential equation?
The general form of a first order differential equation is dy/dx = f(x, y) or M(x, y)dx + N(x, y)dy = 0.
- dy/dx = f(x, y) is called the normal form.
- M(x, y)dx + N(x, y)dy = 0 is useful for checking exactness.
- These forms help classify equations as separable, linear, exact, or homogeneous.
3. How do you solve a separable first order differential equation?
A separable differential equation is solved by separating variables and integrating both sides.
- Step 1: Rewrite as g(y)dy = f(x)dx.
- Step 2: Integrate both sides.
- Step 3: Add constant of integration C.
Integrating: y = x2 + C.
4. What is a linear first order differential equation?
A linear first order differential equation has the form dy/dx + P(x)y = Q(x).
- P(x) and Q(x) are functions of x only.
- The dependent variable y is not squared or multiplied by its derivative.
- It is solved using an integrating factor.
5. What is the integrating factor in a first order linear differential equation?
The integrating factor (IF) for dy/dx + P(x)y = Q(x) is e∫P(x)dx.
- Multiply the whole equation by the integrating factor.
- The left side becomes the derivative of a product.
- Then integrate both sides to find the solution.
6. What is an exact first order differential equation?
An exact differential equation is of the form M(x, y)dx + N(x, y)dy = 0 where ∂M/∂y = ∂N/∂x.
- This condition ensures the equation comes from a potential function.
- The solution is obtained by finding a function F(x, y) such that dF = 0.
- Final solution is written as F(x, y) = C.
7. What is the difference between linear and separable first order differential equations?
The main difference is that a separable equation allows variables to be separated directly, while a linear equation follows the form dy/dx + P(x)y = Q(x).
- Separable form: g(y)dy = f(x)dx.
- Linear form: dy/dx + P(x)y = Q(x).
- All separable linear equations are linear, but not all linear equations are separable.
8. What is the general solution of a first order differential equation?
The general solution of a first order differential equation contains one arbitrary constant C.
- It represents a family of curves.
- Example: For dy/dx = 3x, the general solution is y = 3x2/2 + C.
- A specific value of C gives a particular solution.
9. What is an initial value problem in first order differential equations?
An initial value problem (IVP) is a first order differential equation with a given initial condition like y(x₀) = y₀.
- First find the general solution.
- Substitute the initial condition.
- Solve for the constant C to get the particular solution.
10. Where are first order differential equations used in real life?
First order differential equations are used to model processes involving rates of change such as growth, decay, and motion.
- Exponential growth and decay (population, radioactive decay).
- Newton’s law of cooling.
- Mixing problems in chemistry.
- Simple electrical circuits (RC circuits).
