How to set up differential equations for algebraic manipulation?

In summary, the conversation is about a calculus class where they have been assigned a worksheet on differential equations. The person speaking is having trouble setting up the last three problems on the sheet and is seeking help. They mention that the equations need to be made separable through algebraic manipulation and ask for assistance with problem #18 and #19. The other person offers some tips and hints for approaching these problems and mentions that problem #20 is actually easier than it seems. They also mention that #18 and #19 are not separable but can be solved using integrating factors. The conversation ends with the hope that this information helps the person with their worksheet.
  • #1
wetwilly92
8
0
Hey guys,
We skimmed a chapter on differential eqns in my semester 2 calculus class and we have a worksheet to fill out. I'm having trouble setting up the last 3 problems in this sheet.

http://dl.dropbox.com/u/85600/DIFF_HW.PDF" [Broken]

I'm pretty sure It's all just algebraic manipulation to make the eqns separable, but I'm having trouble "massaging" the eqns into a separable form.

Can anyone help me out?
 
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  • #2
For #18, would you be able to solve
[tex]x \frac{du(x)}{dx} + u(x) = x[/tex]?

For #19, what is d/dx(x² y)?

For #20, the approach looks OK, did you get stuck?
 
  • #3
#20 is easier than you are making it. you don't need to solve it to answer the question. y will be decreasing when dy/dx is negative. you have a formula for dy/dx so you just need to find the intervals on which it is negative.

#'s 18 and 19 are not separable. they are linear and first-order, so integrating factors will work (after a pinch of 'massaging' as you so nicely put it).

hope this helps
 
  • #4
Here's is what is in the link.
 

Attachments

  • DIFF_HW.PDF
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  • Diff HW.jpg
    Diff HW.jpg
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What is a differential?

A differential is a mathematical concept used to describe the rate of change of a function. It is represented by the symbol "d" and is often used in calculus to find the slope of a curve at a specific point.

What is the purpose of a differential?

The purpose of a differential is to measure and describe the rate of change of a function. It allows us to understand how quickly a function is changing at a specific point, and can be used to solve various problems in physics, engineering, and other fields.

What are the basic types of differentials?

The two most commonly used types of differentials are ordinary differentials and partial differentials. Ordinary differentials are used for functions with only one independent variable, while partial differentials are used for functions with multiple independent variables.

What is the difference between a derivative and a differential?

A derivative is the mathematical result of taking the differential of a function. In other words, a derivative is the slope of a function at a specific point, while a differential is the mathematical expression used to represent this slope.

How are differentials used in real-world applications?

Differentials are used in a variety of real-world applications, including physics, engineering, economics, and medicine. They are especially useful for modeling and predicting the behavior of complex systems, such as the stock market or the spread of diseases.

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