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Calculus and Beyond Homework Help
Solving PDE Questions: Constant & Variable Coefficient Equations
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[QUOTE="Tsunoyukami, post: 4510787, member: 274774"] I agree that it is difficult to solve this problem using the geometric method - the important thing is that you understand what's happening in the geometric method. You've rewritten it in the way I hoped you would so let think about that a little bit. What exactly does that expression mean? You're getting there with the co-ordinate method. However, Using the substitutions ##x' = ax + by## and ##y' = bx - ay## as you have we can go even farther. Let's find expression for ##u_{x}## and ##u_{y}## in a nifty way. ##\frac{∂u}{∂x} = \frac{∂u}{∂x'} \frac{∂x'}{∂x} + \frac{∂u}{∂y'} \frac{∂y'}{∂x}## (All we've done is used the chain rule.) Simplify that and do the same thing for ##u_{y}## and plug these new expressions in - you should find you get a much simpler result (it should be on ODE). You can apply the same procedure for the harder PDE you have. [/QUOTE]
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Solving PDE Questions: Constant & Variable Coefficient Equations
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