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I recently came across a system of PDEs $\frac{\partial S}{\partial z}= f_1(x,y,z,w,t)$, $\frac{\partial S}{\partial w}= f_2(x,y,z,w,t)$, $\frac{\partial S}{\partial t}= f_3(x,y,z,w,t)$, $S(x,y,1,1,1)=f_4(x,y)$, where $S$ is an unknown function of five variables $x,y,z,w,t$ and $f_i$ are known. The question is how to obtain a general solution for $S$?

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It seems this is not the right place for your question, as this site is mainly devoted to research matter. Try other sites as or other quoted in the FAQ section, where you may have more chances. – Pietro Majer Aug 30 '10 at 14:16
Pietro: I think that you have been too quick at downvoting this question and its answers. I can imagine research mathematicians in some areas wanting a quick answer on something elementary in a different area. Maybe this ought to be dicussed in meta... – José Figueroa-O'Farrill Aug 30 '10 at 14:20
Hi José, I agree with you in principle, though my feeling in writing the comment was that this particular question could really have more chance of getting an answer in other sites devoted to elementary maths (no answer had been posted while I was wirting the comment). Btw, I didn't vote down the question, nor your answer. – Pietro Majer Aug 30 '10 at 14:41
Pietro: sorry for assuming you had downvoted -- your comment appeared almost simultaneously with the downvotes. I would tend to give the benefit of the doubt in these cases, but perhaps you are right. I have not used the wiki reference desk, but perhaps MU might be a good site for that question. – José Figueroa-O'Farrill Aug 30 '10 at 14:46
I agree with Pietro. For one thing, the overly abstract and contrived formulation makes it sound more like a homework problem than something that arose in research. – Deane Yang Aug 30 '10 at 15:28
up vote 7 down vote accepted

What you have is a family of PDEs labelled by $x,y$ and for fixed $x,y$ you have an equation $$dS = f_1 dz + f_2 dw + f_3 dt$$ for a function $S:\mathbb{R}^3 \to \mathbb{R}$ of the three variables $z,w,t$. The first thing to check is that the equation is integrable: namely, that the 2-form $d(dS) = 0$. If that is the case, then $S$ exists up to a locally constant function, which you fix from the fourth condition.

To find the explicit solution, choose any path $\gamma$ from $(1,1,1)$ to $(z,w,t)$ and integrate the right-hand side of the equation: $$S(x,y,z,w,t) = S(x,y,1,1,1) + \int_\gamma (f_1 dz + f_2 dw + f_3 dt).$$

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By using method of characteristics. See the book of Zachmanoglous Thoe " Introduction to Partial differential equations with applications" or Evans "Partial differential equations" GSM 19 AMS

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