First-Order Equations: don't understand a simplification step

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In summary: The reason this is important is that the product rule tells us that the derivative of a product is the first function times the derivative of the second, plus the second function times the derivative of the first. By recognizing the left side as the derivative of a product, we can rewrite the equation in a form that will be easier to integrate.
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There is a specific step when solving first-order equations introduced in my textbook when describing the process for solving with an integrating factor, and it has been popping up everywhere - I do not understand how it works and the book doesn't explain it well.

Example 1:

x(dy/dx) + y = sin(x) / x

becomes

d/dx(xy) = sin(x) / x

Example 2:

e^-x (dy/dx) - y(e^-x) = e^(-4x/3)

becomes

d/dx(y(e^-x)) = e^(-4x/3)I'm having trouble rationalizing how this step is performed. Thanks for any help!
 
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Kavorka said:
There is a specific step when solving first-order equations introduced in my textbook when describing the process for solving with an integrating factor, and it has been popping up everywhere - I do not understand how it works and the book doesn't explain it well.

Example 1:

x(dy/dx) + y = sin(x) / x

becomes

d/dx(xy) = sin(x) / x

Example 2:

e^-x (dy/dx) - y(e^-x) = e^(-4x/3)

becomes

d/dx(y(e^-x)) = e^(-4x/3)I'm having trouble rationalizing how this step is performed. Thanks for any help!
In both examples they are recognizing that the left side is the derivative, with respect to x, of a product. In the first example, the product is xy. In the second, the product is ye-x.
 

1. What is a first-order equation?

A first-order equation is a mathematical equation that only contains the first power of the variable. It can be in the form of y = mx + b, where m and b are constants, or in the form of dy/dx = f(x), where f(x) is a function.

2. What is the purpose of simplifying a first-order equation?

Simplifying a first-order equation helps to make it easier to solve or analyze. It reduces the number of terms and makes the equation more manageable and understandable.

3. How do I know which simplification steps to take?

The simplification steps will depend on the specific equation you are working with. However, some common steps include combining like terms, factoring, and using the distributive property. It is important to follow the order of operations and simplify one step at a time.

4. What should I do if I don't understand a simplification step?

If you are having trouble understanding a simplification step, you can try breaking the step down into smaller parts or asking for clarification from a teacher or tutor. You can also try looking up examples or explanations online or in a textbook.

5. Can I skip simplification steps?

No, it is important to follow all the necessary simplification steps in order to solve the equation correctly. Skipping steps may result in an incorrect solution or make the problem more difficult to solve.

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