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## Differential Equations Homework Help

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## How can MEB help you with differential equations homework?

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### About differential equations?

#### What are differential equations?

Differential equations are mathematical statements that connect functions to their derivatives. These derivatives express the rates of change of quantities whose values are continuously changing. In real-world situations, these functions stand for physical quantities, and the derivatives denote the rate of change of these physical quantities.

The relationship between the physical quantities and their variation rates is common, and differential equations capture them very well. Differential equations are particularly important in engineering, science, and many other fields that study quantitative changes. They help us directly observe and measure essential changes in these systems. Many vital fields like physics, engineering, biology, and economics extensively use differential equations.

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#### Differential equations and calculus

Calculus is a fundamental branch of mathematics that deals with change. Derivatives represent the rates at which variables change. Differential equations are a part of calculus that allows us to use an unknown function and its derivative and set up an equation containing both of them. The equations predict the change in the variables and offer insight into how and why they change.

#### Why are differential equations important?

Differential equations are helpful in many subjects, from economics to engineering, including physics, chemistry, and biology.

Differential equations come in handy in predicting the outcomes of real-life situations with high accuracy. Here are some examples from our daily lives where they are commonly put to use.

- Medicine – modeling the spread of diseases or cancer growth
- Ecology – population growth of species, conservation biology
- Physics – describe the motion of pendulums, waves, or chaotic systems
- Engineering – fluid mechanics, circuit design, and heat transfer
- Finance – continuous compound interest problems
- Chemistry – calculate radioactive half-life and model chemical reactions
- Optimizing techniques – techniques like expectation-maximization or gradient descent

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#### Order of differential equations

The highest order derivative in a differential equation is the order of the equation. For example, the equation: dy/dx = f (x,y) is a function of two variables x and y. The highest differential is of the first order. Hence, it is a first-order differential equation.

#### Degree of differential equations

The power of the highest order derivative of the differential equation is the degree of the equation. We arrive at it after rationalizing and integrating all the derivatives in the equation.

#### Types of differential equations

- Ordinary differential equations
- Linear differential equations
- Nonlinear differential equations
- Partial differential equations
- Homogeneous differential equations
- Nonhomogeneous differential equations

#### Differential equations solutions

The solution for a differential equation is a function that satisfies the equation. A general solution contains as many arbitrary constants as the differential equation order. On the other hand, a particular solution is free from arbitrary constants, as the name suggests.

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#### Differential equations topics

Below is a list of some of the most important topics in a differential equations course.

- Bessel function of the first kind
- Euler forward method
- Fourier transform
- Laplace transform
- Separation of variables
- Slope field

Watch this youtube video on differential equations if you like learning by watching videos (My Engineering Buddy is not associated with this video)

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