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Solving PDE Questions: Constant & Variable Coefficient Equations
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[QUOTE="Tsunoyukami, post: 4510725, member: 274774"] In general PDEs are much more difficult to solve than ODEs so in both of these problems I think it prudent to try to reduce them into ODEs using a few "tricks". I agree that your first question looks more daunting initially but we'll get to that as soon as we work through the second question (because it's easier and I think it's more enlightening). Before we do that though, we can try to dumb things down and solve some PDEs that are much simpler and work our way into solving these slightly harder ones. 1) First, what is the general solution to the PDE ##u_{x} = 0## where u is a function of x and y? 2) Second, what is the general solution to the PDE ##au_{x} + bu_{y} = 0## where u is a function of x and y and a and b are constants not both zero? This is a simple extension of my first question but it is illuminating. Hint: Think about another way you could represent ##au_{x} + bu_{y}## using methods from multivariable calculus (Hint: Think about the gradient.). Now that you (hopefully) have the general solution of ##au_{x} + bu_{y} = 0## think about your second question: ##au_{x} + bu_{y} = 1##. What does this [I]mean[/I] geometrically?If you're interested in a procedure of attack for solving these problems there are two methods that I know about. One is called the geometric method (this is the method I am trying to lead you to by asking my above questions). The second is the coordinate method which I personally find works [I]very[/I] nicely (especially for problem of only 2 variables). You might be able to find these in your textbook. In fact, you can solve both problems using this method but understanding the geometric method will really help you understand these types of PDEs better. Hopefully this helps! [/QUOTE]
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Solving PDE Questions: Constant & Variable Coefficient Equations
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