First Order Differential Equations

A first-order differential equation is one of the fundamental types of differential equations encountered in calculus and mathematical modeling. In this blog, we will explore what first-order differential equations are, how to solve them, and provide a clear example of a first-order differential equation solution. 

A First-Order Differential Equation involves a function and its first derivative, typically in the form $\frac{dy}{dx}=f(x,y)$. The goal is to find y(x) that satisfies this equation. Types include separable, linear, exact, and homogeneous equations, each requiring different solution methods. First-order differential equations model many real-world phenomena, such as population growth, heat transfer, and motion, making their study essential in fields like physics, engineering, and economics.

1.0What Is A First-Order Differential Equation?

A first-order differential equation is an equation that involves a function and its first derivative but no higher derivatives. In mathematical terms, it is usually expressed as:$\frac{dy}{dx}=f(x,y)$

Where:

• y is the dependent variable (the function),
• x is the independent variable, and
• f (x, y) is a function of both x and y.

The "first-order" part of the name comes from the fact that the highest derivative in the equation is the first derivative of y with respect to x.

Example of a First-Order Differential Equation

A simple example of a first-order differential equation is:$\frac{dy}{dx}=3x^2+2y$

In this case, the equation involves the first derivative of y with respect to x, and it has terms that depend on both x and y.

2.0Methods Of Solving A First-Order Differential Equation

To solve a first-order differential equation, there are different techniques depending on the structure of the equation. Some common methods for solving first-order differential equations include:

• Separation of Variables
• Integrating Factor Method
• Homogeneous Equations
• Exact Equations

3.0Solve First-Order Differential Equation By Separation Of Variables

Consider the following first-order differential equation:$\frac{dy}{dx}=\frac{x}{y}$

We’ll solve this equation using the separation of variables method.

Step 1: Rearrange the equation

We start by rewriting the equation in a form where all terms involving y are on one side and all terms involving x are on the other side: $ydy=6xdx$

Step 2: Integrate both sides

Now, we perform integration on both sides of the equation with respect to their respective variables:$\int ydy=\int 6xdx$

On the left-hand side, the integral of y dy is $\frac{y^2}{2}$, and on the right-hand side, the integral of 6x dx is $3x^2$. Thus, we have: $\frac{y^2}{2}=3x^2+C$

Where C is the constant of integration.

Step 3: Solve for y

To find the solution for y, we multiply both sides of the equation by 2: $y^2=6x^2+2C$

Finally, taking the square root of both sides, we get: $y=\pm \sqrt{6x^2+2C}$

This is the general solution of the first-order differential equation.

4.0Solving a First-Order Linear Differential Equation Using The Integrating Factor (IF)

1. Write the equation in standard form: Ensure that the equation is in the form . $\frac{dy}{dx}+P(x)y=Q(x)$
2. Find the Integrating Factor (IF): The integrating factor is given by: $\mu (x)=e^{\int P(x)dx}$

The integrating factor is a function that, when multiplied with both sides of the equation, allows you to simplify and solve it.

3. Multiply through by the integrating factor: Multiply the entire differential equation by $\mu (x)$, which will make the left-hand side of the equation the derivative of $\mu (x)y$.
4. Integrate both sides: After multiplying by the integrating factor, integrate both sides with respect to x.
5. Solve for y(x): After performing the integration, solve for the function y(x).

Example:  $\frac{dy}{dx}+2=4x$

Step 1: Write the equation in standard form

The given equation is already in standard form:

$\frac{dy}{dx}+P(x)y=Q(x)$

where P(x) = 2 and Q(x) = 4x.

Step 2: Find the Integrating Factor (IF)

The integrating factor is calculated as:

 $\mu (x)=e^{\int P(x)dx}=e^{\int 2dx}=e^{2x}$

Step 3: Multiply through by the integrating factor

Now, multiply the entire equation by $\mu (x)=e^{2x}$:

$e^{2x}.\frac{dy}{dx}+e^{2x}.2y=e^{2x}.4x$

This simplifies to:

$\frac{d}{dx}(e^{2x}y)=4x{e}^{2x}$

Notice that the left-hand side is the derivative of $e^{2x}y$with respect to x.

Step 4: Integrate both sides

Next, integrate both sides with respect to x:

$\int \frac{d}{dx}(e^{2x}y)dx=\int 4x{e}^{2x}dx$

The left-hand side integrates to: $e^{2x}y$

For the right-hand side, we need to use integration by parts to solve $\int 4x{e}^{2x}dx$. The formula for integration by parts is: 

$\int udv=uv-\int vdu$

$\int 4x{e}^{2x}dx=4x.\frac{1}{2}{e}^{2x}-\int \frac{1}{2}{e}^{2x}.4dx$

$=2x{e}^{2x}-2\int e^{2x}dx$

$=2x{e}^{2x}-e^{2x}+C$

Now, equating both sides:

$e^{2x}y=2x{e}^{2x}-e^{2x}+C$

Step 5: Solve for y(x) 

Finally, divide both sides of the equation by $e^{2x}$ to solve for y:

$y=2x-1+C{e}^{-2x}$

This is the general solution to the differential equation.

5.0Sample Questions on First-Order Differential Equation

1. What is a separable differential equation? 

Ans: A separable equation is one that can be written as $\frac{dy}{dx}=g(x)h(y)$, allowing variables to be separated and integrated.

2. What is an example of a linear first-order differential equation? 

Ans: A linear first-order equation has the form $\frac{dy}{dx}+P(x)y=Q(x)$, where P(x) and Q(x) are functions of x.

3. How do you recognize an exact equation? 

Ans: An equation is exact if it can be written as M(x, y)dx + N(x, y)dy = 0 and satisfies the condition $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$.