First-Order Differential Equation

First-Order Differential Equation: First-order differential equations are defined by an equation dy/dx =f (x, y) where x and y are two variables and f(x, y) are two functions. It is defined as a region in the xy plane. These types of equations have only the first derivative dy/dx so that the equation is of the first order and no higher-order derivatives exist.

Differential equations of the first order are written as;

y’ = f (x,my)

(d/dx)y = f (x, y)

Let’s learn more about First-order Differential Equations, types, and examples of First-order Differential equations in detail below.

What is a First-Order Differential Equation?

A first-order differential equation is defined by an equation: dy/dx = f(x, y) involving two variables x and y, where the function f(x,y) is defined on a region in the xy-plane. For any linear expression in y, the first-order differential equation [Tex]y’ = f (x, y)[/Tex] is linear. Nonlinear differential equations are those that aren’t linear.

Example of First-Order Differential Equation

Some examples of first-order differential equation

• dy/dx = 2x
• dy/dx = x + 11
• dy/dx = 4x – 5

This equation represents a first-order ordinary differential equation where the derivative of y concerning x is equal to 2x.

Types of First-Order Differential Equations

First-order differential Equations are classified into several forms, each having its characteristics. Types of the First-Order Differential Equations:

• Linear Differential Equations
• Homogeneous Equations
• Exact Equations
• Separable Equations
• Integrating Factor

Linear Differential Equations

A linear differential equation consists of a variable, its derivative, and additional functions. It’s expressed in the standard form as:

dy/dx + P(x)y = Q(x)

where,

• P(x) and Q(x) may be functions of x or numerical constants.

Homogenous First Order Differential Equation

A homogeneous differential equation is a function of the form (f(x,y) \frac{dy}{dx} = g(x,y)), where the degree of (f) and (g) is the same. A function (F(x,y)) can be considered homogeneous if it satisfies the condition: (F(\lambda x, \lambda y) = \lambda^n F(x,y)) for any nonzero constant (\lambda).

• In simpler terms, a differential equation is homogeneous if it involves a function and its derivatives in a way that maintains a consistent degree.

Example of Homogenous First Order Differential Equation

Consider the differential equation: (\frac{dy}{dx} = \frac{x^2 – y^2}{xy}). This equation is homogeneous because both the numerator and denominator have the same degree (1).

Exact Differential Equations

The formula Q (x,y) dy + P (x,y) dx = 0 is considered to be an exact differential equation if a function f of two variables, x and y, exists that has continuous partial derivatives and can be divided into the following categories.

The general solution of the equation is:

u(x, y) = C since ux(x, y) = p(x, y) and uy (x, y) = Q(x, y)

where,

• C is Constant of Integration

Separable Differential Equations

Separable differential equations are a special type of differential equations where the variables involved can be separated to find the solution of the equation. Separable differential equations can be written in the form of:

dy/dx = f(x) × g(y) where x and y are the variables and are explicitly separated from each other.

Once the variables have been separated, it is simple to find the differential equation’s solution by integrating both sides of the equation. After the variables are separated, the separable differential equation

dy/dx = f(x) × g(y) is expressed as dy/g(y) = f(x) × dx

Integrating Factor Homogenous Differential Equation

The integrating factor is a function used to solve first-order differential equations. It is most commonly applied to ordinary linear differential equations of the first order. If a linear differential equation is written in the standard form y’ + a(x)y = 0

Then, the integrating factor (μ) is defined as: (\mu = e^{\int P(x)dx})

First-Order Differential Equation Solution

First-Order Differential Equation is generally solved and simplified using two methods mentioned below.

• Using Integrating Factor
• Method of variation of constant

Solution using Integration Factor

Using integrating factor can be used to simplify and facilitate the solution of linear differential equations. The entire equation becomes exact when the integrating factor, which is a function of x, is multiplied.

For any linear differential equation is written in the standard form as:

y’ + a(x)y = 0

Then, the integrating factor is defined as:

u(x) = e^(∫a(x)dx)

Multiplication of integrating factor u(x) to the left side of the equation converts the left side into the derivative of the product y(x).u(x). General solution of the differential equation is:

y = {∫u(x).f(x)dx + c}/u(x)

where C is an arbitrary constant.

Method of Variation of a Constant

Method of Variation of a Constant is a similar method to solve first order differential equation. In first step, we need to do y’ + a(x)y = 0. In this method of solving first order differential equation, homogeneous equation always contains a constant of integration C.

• In solving non-homogeneous linear differential equations, we replace the constant (C) with an unknown function (C(x)).
• By substituting this solution into the non-homogeneous equation, we can determine the specific form of the function (C(x)).
• This approach allows us to find a particular solution that satisfies the given non-homogeneous equation.
• Interestingly, both the method of variation of constants and the method of integrating factors lead to the same solution.

Remember, this technique helps us handle non-homogeneous differential equations by introducing a function that varies with the independent variable!

Properties of First-Order Differential Equations

Various properties of linear first-order differential equation are:

• No transcendental functions like trigonometric functions and logarithmic functions are used in linear first-order differential equation.
• In linear first-order differential equation, products of y and any of its derivatives are not present.

First-Order Differential Equation Formulas

[Tex]\begin{array}{|c|c|c|} \hline \textbf{Type of Equation} & \textbf{General Form} & \textbf{Solution Method} \\ \hline \text{Linear Differential Equations} & \frac{dy}{dx} + p(x)y = q(x) & \text{Use an integrating factor, } \mu(x) = e^{\int p(x) \, dx}, \text{ then solve } \frac{d}{dx}(\mu(x)y) = \mu(x)q(x). \\ \hline \text{Homogeneous Equations} & \frac{dy}{dx} = f\left(\frac{y}{x}\right) & \text{Make the substitution } v = \frac{y}{x}, \text{ then solve the resulting separable differential equation for } v. \\ \hline \text{Exact Equations} & M(x,y) + N(x,y)\frac{dy}{dx} = 0 & \text{Find a potential function } \Psi \text{ such that } d\Psi = Mdx + Ndy, \text{ where } \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}. \\ \hline \text{Separable Equations} & \frac{dy}{dx} = g(x)h(y) & \text{Separate variables and integrate: } \int \frac{1}{h(y)} dy = \int g(x) dx + C. \\ \hline \text{Integrating Factor} & \text{Often used for non-exact linear equations} & \text{Find an integrating factor } \mu(x) \text{ that makes the equation exact, then proceed as with exact equations.} \\ \hline \end{array} [/Tex]

Applications of First-Order Differential Equations

Numerous disciplines, including physics, engineering, biology, economics, and more, first-order differential equations are used. Among other things, they are used to simulate phenomena like fluid dynamics, electrical circuits, population dynamics, and chemical reactions. Various applications of the first-order differential equation are:

• Used in Newton’s Law of Cooling
• Used in Orthogonal Trajectories
• Used in Falling Body Problems
• Used in Dilution Problems

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First Order Differential Equation Examples and Solutions

Below are the example of problems on First Order Differential Equation.

Example 1: Solve the following separable differential equation:dy/dx = x/y²

Solution:

First, we separate the variables:

y^2.dy = x.dx

Integrating both sides:

∫y^2.dy = ∫x.dx

y³/3 = x²/2 + C

Example 2: Solve the following linear differential equation: dy/dx + 2xy – x = 0

Solution:

Equation in the standard form:

dy/dx + 2xy – x = 0

Now, we can use an integrating factor to solve it:

f(x) = e ^∫2xdx

f(x) = [Tex]e^{x^2}[/Tex]

Multiplying both sides by the integrating factor:

[Tex]e^{x^2} \frac{dy}{dx} + 2xye^{x^2} – xe^{x^2} = 0[/Tex]

[Tex]\frac{d}{dx} (ye^{x^2}) – xe^{x^2} = 0[/Tex]

Integrating both sides:

[Tex]ye^{x^2} = \frac{x^2}{2} + C[/Tex]

Example 3: Solve the first-order differential equation x³y’ = x + 2

Solution: 

x³y’ = x + 2

⇒ y’ = (x + 2)/x³

Integrating both sides w.r.t. x

⇒ ∫(dy/dx) dx = ∫ {(x + 2)/x³} dx

⇒ y = -1/x – 1/x² + C

First Order Differential Equation Questions

Q1: For differential equation dy/dx + yx² = sin x find integrating factor.

Q2: Find the solution of differential equation dy/dx = y²(x²+1).

Q3: Solve the differential equation dy/dx + 2x³y = x.

FAQs on First-Order Differential Equation

What is First Order Differential Equation?

A first order differential equation is a differential equation where the maximum order of a derivative is one and no other higher-order derivative can appear in this equation. A first-order differential equation is generally of the form [Tex]dy/dx =f (x,y)[/Tex].

What are the types of First Order Differential Equations?

First Order Differential Equations are classified into three categories: (i) Separable Differential Equation, (ii) Linear Differential Equation and (iii) Exact Differential Equation

Give one example of First Order Differential Equation.

An Example of First Order Differential Equations: (dy/dx = 2x)

What are application of First Order Differential Equations?

First Order Differential Equation are used in various fields like physics, engineering, biology, applied mathematics, etc.

What is Homogeneous First Order Differential Equation?

A first order differential equation M(x, y) dx + N(x, y) dy = 0 is said to be homogeneous if both M(x, y) and N(x, y) are homogeneous. We can write a homogeneous linear first-order differential equation is of the form y’ + p(x)y = 0.