Differential Equation and Slope Field Questions.by PFStudent Tags: differential, equation, field, slope 

#1
Apr1409, 08:32 PM

P: 171

Hey,
1. The problem statement, all variables and given/known data. I was reading through the Differential Equation portion of my textbook and didn't quite understand the following paragraph. [tex] {y} = {f(x)} [/tex] Where, [tex] {{\frac {{d}^{}}{d{x}^{}}}{\Big[y\Big]}} = {{{f}^{\prime}}{(x)}} [/tex] so that it is understood that [itex]{{{f}^{\prime}}{(x)}}[/itex] like [itex]{f(x)}[/itex] is a function of only [itex]{x}[/itex]. However, in the above paragraph it is insisted that [itex]{{\frac {{d}^{}}{d{x}^{}}}{\Big[y\Big]}}[/itex] must be a function of [itex]{x}[/itex] and [itex]{y}[/itex]. In other words [itex]{{y}^{\prime}} = {{F}{(x, y)}}[/itex], why is that? 2. Relevant equations. Knowledge Differential Equations and Slope Fields. 3. The attempt at a solution. If we begin from the conventional notation that, [tex] {z} = {f(x, y)} [/tex] I run in to the problem that I don't know how to explicitly find [itex]{{z}^{\prime}}[/itex]. Specifically, I don't know how to differentiate [itex]{f(x, y)}[/itex] with respect to [itex]{x}[/itex] and [itex]{y}[/itex] simultaneously. So that the derivative is actually a function of [itex]{x}[/itex] and [itex]{y}[/itex], like [itex]{f(x, y)}[/itex]. How would I differentiate [itex]{z}[/itex] with respect to [itex]{x}[/itex] and [itex]{y}[/itex] simultaneously? I note however, that in this particular case we're talking about [itex]{{y}^{\prime}}[/itex] as opposed to [itex]{{z}^{\prime}}[/itex]. Noting this I recall that we can rewrite, [tex] {\frac {dy}{dx}} = {{{f}^{\prime}}{(x)}} [/tex] as [tex] {dy} = {{{{f}^{\prime}}{(x)}}{dx}} [/tex] Where [itex]{y}[/itex] is found by integrating both sides of the above equation. However, when I try to do this with the equation given, [tex] {{y}^{\prime}} = {F(x, y)} [/tex] Which can be rewritten as, [tex] {dy} = {F(x, y)dx} [/tex] and when integrated is, [tex] {y} = {{\int_{}^{}}{F(x, y)dx}} [/tex] I find that I do not know how to evaluate the RHS. How would I evaluate it? Thanks, PFStudent 



#2
Apr1509, 07:40 PM

P: 171




#3
Apr1509, 07:41 PM

P: 25





#4
Apr1609, 09:30 AM

P: 171

Differential Equation and Slope Field Questions. 



#5
Apr1609, 10:04 AM

P: 54





#6
Apr1609, 02:48 PM

P: 25

what text book are you using?



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