How do you solve a first-order differential equation?

The method of solution depends on the type of equation, such as separation of variables, integrating factor (for linear equations), or using exact equations.

Can first-order differential equations have multiple solutions?

Yes, depending on the initial conditions or boundary values, there can be one or multiple solutions to a first-order differential equation.

What is the integrating factor method?

The integrating factor method is used for linear first-order differential equations to make the equation easier to solve by multiplying both sides by a specific function.

What is the solution to a homogeneous first-order differential equation?

A homogeneous equation can be solved using substitution methods or by reducing it to a separable equation.

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First Order Differential Equations

Q: What is a first-order differential equation?

A first-order differential equation involves the first derivative of a function and represents how the function changes with respect to one variable.

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 $\frac{d y}{d x} = f (x, y)$ . 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: $\frac{d y}{d x} = f (x, y)$

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: $\frac{d y}{d x} = 3 x^{2} + 2 y$

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: $\frac{d y}{d x} = \frac{x}{y}$

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: $y d y = 6 x d x$

Step 2: Integrate both sides

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

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 $3 x^{2}$ . Thus, we have: $\frac{y ^{2}}{2} = 3 x^{2} + C$

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: $y^{2} = 6 x^{2} + 2 C$

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

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

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

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

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

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

Example: $\frac{d y}{d x} + 2 = 4 x$

Step 1: Write the equation in standard form

The given equation is already in standard form:

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

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

Step 2: Find the Integrating Factor (IF)

The integrating factor is calculated as:

$μ (x) = e^{\int P (x) d x} = e^{\int 2 d x} = e^{2 x}$

Step 3: Multiply through by the integrating factor

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

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

This simplifies to:

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

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

Step 4: Integrate both sides

Next, integrate both sides with respect to x:

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

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

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

$\int u d v = uv - \int v d u$

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

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

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

Now, equating both sides:

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

Step 5: Solve for y(x)

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

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

This is the general solution to the differential equation.

5.0Sample Questions on First-Order Differential Equation

What is a separable differential equation?

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

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

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

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 $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ .

Also Read:

Exact Differential Equation	Methods of Differentiation	Differentiation
important integration formulas for jee	Differentiability	Integral Calculus
Riemann Integral	Partial Differential Equations	Differentiation and Integration

Frequently Asked Questions

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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 $\frac{d y}{d x} = f (x, y)$ . 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: $\frac{d y}{d x} = f (x, y)$

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: $\frac{d y}{d x} = 3 x^{2} + 2 y$

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: $\frac{d y}{d x} = \frac{x}{y}$

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: $y d y = 6 x d x$

Step 2: Integrate both sides

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

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 $3 x^{2}$ . Thus, we have: $\frac{y ^{2}}{2} = 3 x^{2} + C$

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: $y^{2} = 6 x^{2} + 2 C$

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

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

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

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

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

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

Example: $\frac{d y}{d x} + 2 = 4 x$

Step 1: Write the equation in standard form

The given equation is already in standard form:

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

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

Step 2: Find the Integrating Factor (IF)

The integrating factor is calculated as:

$μ (x) = e^{\int P (x) d x} = e^{\int 2 d x} = e^{2 x}$

Step 3: Multiply through by the integrating factor

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

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

This simplifies to:

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

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

Step 4: Integrate both sides

Next, integrate both sides with respect to x:

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

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

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

$\int u d v = uv - \int v d u$

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

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

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

Now, equating both sides:

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

Step 5: Solve for y(x)

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

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

This is the general solution to the differential equation.

5.0Sample Questions on First-Order Differential Equation

What is a separable differential equation?

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

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

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

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 $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ .

Also Read:

Exact Differential Equation	Methods of Differentiation	Differentiation
important integration formulas for jee	Differentiability	Integral Calculus
Riemann Integral	Partial Differential Equations	Differentiation and Integration

Frequently Asked Questions

What is a first-order differential equation?

How do you solve a first-order differential equation?

Can first-order differential equations have multiple solutions?

What is the integrating factor method?

What is the solution to a homogeneous first-order differential equation?

Join ALLEN!

First Order Differential Equations

1.0What Is A First-Order Differential Equation?

2.0Methods Of Solving A First-Order Differential Equation

3.0Solve First-Order Differential Equation By Separation Of Variables

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

5.0Sample Questions on First-Order Differential Equation

Table of Contents

Frequently Asked Questions

What is a first-order differential equation?

How do you solve a first-order differential equation?

Can first-order differential equations have multiple solutions?

What is the integrating factor method?

What is the solution to a homogeneous first-order differential equation?

Join ALLEN!

First Order Differential Equations

1.0What Is A First-Order Differential Equation?

2.0Methods Of Solving A First-Order Differential Equation

3.0Solve First-Order Differential Equation By Separation Of Variables

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

5.0Sample Questions on First-Order Differential Equation

Table of Contents